If Bob Crow were a business leader, he would probably be celebrated as a great British success story. Since taking charge in 2002 he has increased the profits of his shareholders year on year, even through the most testing of economic times, while expanding his business by 50%. By nearly any measure you care to choose, Crow is far and away the most successful leader in his field. And yet he is widely regarded as a national disaster.

Before meeting the RMT union boss, I asked some friends what they thought of him. The bloke's "a nutter and a thug", they all said – a dinosaur, the unacceptable face of socialism – and these were people who consider themselves leftwing. The right is much more hostile. Boris Johnson has described Crow as "demented", the London Evening Standard called him the Most Hated Man in London, and the Sun blockaded his home with a double-decker bus. If press coverage of RMT's current dispute with London Underground is to be believed, when Crow brought the capital to a halt again last month, barely a single commuter could be found who supported the tube strike, or even understood what it was about.

Crow greets me in his office at the RMT's London headquarters, looking immaculately groomed if slightly gaudy in a diamond-patterned cardigan and tie. He presents his case for the strike, explaining all the reasons why London Underground shouldn't be allowed to get rid of 800 ticket office staff, which range from customer service standards and the needs of tourists, to safety procedures in the event of a terrorist attack. He talks calmly for at least five minutes without pausing, and it all sounds eminently reasonable. But if the strike is perfectly justified, he must ask himself where he is going wrong, because so few people appear to agree?

"But people must see the news," he says, "and it's been pretty clear what it's about. Job losses and inadequate staffing." I suspect most people see the news and just think, oh God, here we go again. They don't care about the details, because the RMT always seems to be on strike – and they can't see what makes its members so special, when their own jobs are at risk as well.

"Well I'm not being arrogant," he says, quick as a flash, "but why don't they do something about that then?"

Crow has a pretty uncomplicated view of the role of a trade union leader. "Our organisation is purely to look after our members. It's not our job to run the railway network, our job is to represent our members."

Does he have any responsibility to anyone else?

"No." No responsibility whatsoever to the travelling public? "No."

So who does?

"London Underground. They're the management."

I ask if he knows how many strikes he has called since taking over the RMT.

"Nah."

It's reported to be more than any other union leader in any other industry. "Probably, yeah," he agrees. Does he consider that a mark of his success or of his failure?

"I couldn't care less if we had no strikes in 10 years, or we had a million strikes," he says mildly. "Our members vote in a secret ballot, and I respect their wishes."

I'm not sure his indifference can be entirely authentic, because Crow's whole argument is that his union is successful because it is willing to strike. "People join us on the basis that we're prepared to have a fight. We might not always win, but they do like someone having a go. Now our job, by the way, is to get what we want without going on strike. Our members don't get up in the morning going: 'What can we have a strike about?' But our brand is that we're out there, punching away."

Since he took charge in 2002 the RMT's membership has grown from 54,000 to 80,000, and has enjoyed substantial annual pay rises, improved conditions, and even the reopening of a final salary pension scheme. "The Evening Standard had it right, it said I was 'obsessed' with improving my members' living standards. Dead right, I actually get pleasure when I see one of my members get a pay rise. That's another one we've had over them. Yeah, I admit to that." And they get it, according to Crow, because unlike most modern unions they are willing to strike.

"Why was it that in 1978 we had 12 and a half million union members in this country compared to now where we've got six and a half million? Why did people join in 78? Cos the unions had teeth. The last few years you wouldn't even have known the TUC [conference] was on, but the TUC this year was probably the best publicised TV event in 15 or 20 years. Why was it? Cos they was talking about taking action, defending their members and so on. So it demonstrates the fact that if you've got teeth, and you're a force to be reckoned with, then people join you cos they see you can do something."

There is an unarguable logic to Crow's position. In many ways it has a clarity absent from much of the modern trade union movement, which is somehow expected to represent the interests of employers and customers as much as those of its members, while companies still put their shareholders first. In fact, you could say he's the trade union equivalent of Ryanair boss Michael O'Leary – though he looks appalled when I say so – and most of us might secretly like to have someone like Crow fighting our corner. In an ideal world, he says, every trade union would be run by someone like him. But as a guiding political philosophy, his position has two obvious problems – the first of which is that people who work on our railways are not, unlike most of us, competing in a globalised market.

"It's not the same playing field, I will accept. Working on the railway compared to working in a call centre." So how does a trade union help its members, if it insists on pay and conditions that persuade employers to outsource the job to cheaper staff overseas?

"Yep, that's a very, very good argument and I accept what you're saying cos it's the truth. Quite clearly globalisation is about lowering pay and conditions. So therefore our job is to organise globally. That's what we do with the International Transport Workers' Federation. To organise globally to try to raise wages and help unions in other parts of the world for increasing their pay."

Really? If the trade union movement hasn't managed that in this country, I'm not sure how he thinks it's going to achieve it for workers in, say, Mumbai.

"But see, what globalisation's about at the end of the day, it could be a fantastic thing. It could raise wages to the very top. That's what it's going to boil down to. Either we're going to be pushed to the bottom, or pushed to the top. Whoever's strongest is going to get their way. If there's not a strong trade union movement, the employers will get their way. They will drive pay and conditions down for the workers in this country, so who else is going to stand up for working people in this country except the trade union movement? Who else?"

The second problem is not unrelated to his last answer, and might be best described as myopic idealism. If Crow doesn't like the look of something, he doesn't seem to see it. For example, he supports Millwall, a club with a well-documented record of hooliganism, yet he has insisted: "I don't care what people say, I don't see aggravation." Similarly, he has a Staffordshire bull terrier – a notoriously volatile breed – of whom he has said, "They've got a bad name, I know, but he's as soft as anything. If someone broke in he'd lick them to death," which is not a promise I'd ever be tempted to test.

More importantly, he says he cannot think of a single example of an unjustified strike in the last 40 years. Not one. He ticks them off one by one, nodding "totally justified", all the way back to the miners' strike of 1972. So when Ed Miliband talks about irresponsible strike action? He looks at me blankly. "I don't know what he's on about."

Like many people with a tough reputation, Crow can come across as surprisingly sentimental. He was born in east London in 1961, the son of a docker, and began working on the Underground at 16. Though he now lives in a big house in Essex with his partner – they have four children between them – he retains a nostalgia for the old East End.

"I think the 70s was a great time," he says. "I've got to say, and this is God's honest truth, people say they were bad times but I think they was fantastic times. Sunday afternoon, for example – everyone had their dinner at the same time, half-past two, everywhere you walked round east London all you could smell was roast beef and Yorkshire pudding, everyone's old man was down the pub having a drink Sunday lunchtime, every kid was on the wall outside the pub with a bottle of Coke and packet of crisps waiting for their old man to come out the pub. There was jobs everywhere, people would come out of one job and into another. We had the big match on Sunday afternoon, and everyone was happy."

And when rubbish and bodies were piling up in the streets? "Aoowh," he scoffs dismissively, as if this were an absurd detail to mention. "Well look, there's bodies piling up in the streets now, in the sense that there's pensioners scared to turn their heating on."

Crow calls himself a Marxist, and as such is typically maddening. A lot of what he says makes sense – but then he goes and spoils it by saying something that bears almost no resemblance to reality. His commitment to the working class is passionate, but when I ask him to define the working class, he says, "Those who have to go to work and sell their labour to their employer," which would apply to practically everyone, including multi-millionaire bankers. As he himself earns £133,183 as year – "I'm working-class, absolutely" – you can't help wondering if this might explain the breadth of his definition.

We are, he says, living in politically promising times, with a new generation becoming radicalised by the cuts. He seems particularly excited by the student protesters. "As Marx says, capitalism is sowing the seeds of its own destruction, because as technology's come in and put people out of work, it's also giving these young kids the power to organise themselves through the internet."

And he certainly wants to be involved. Officially, only 10% of his job is dedicated to political activism, "But in reality I imagine I'm doing 30% political," and he is inundated with media requests. As his press officer points out, "Bob's in demand cos he's got something interesting to say. You couldn't say that about all these trade union leaders who don't say anything interesting, could you?"

Crow does say interesting things. "I'm not like one of these leftwing commentators who says: 'Oh, don't worry about the deficit.' I'd put a 1 pence tax on every text message that's sent in Britain, that would nearly wipe out half the deficit." He'd also put a windfall tax on every share cashed in. But then he adds: "I'd have no problem – and this is what Denis Healey said, and he was no radical – to squeeze the rich 'til the pips squeak."

What? That comment was a political catastrophe for Healey. As he must surely know. Does Crow actually want to win people round, or not?

"Well it would be nice to walk down the road and all the shutters and windows opened and people started throwing roses out, shouting: 'Morning Bob, how are you?' That would be nice. However, you've got to recognise that the job you do ain't about being nice. The job we do is about defending our members. And as far as I'm concerned, if I can get job security and decent pay for my members I couldn't give two hoots about being unpopular."

Crow's problem seems to be that the very qualities that make him a formidable union leader don't translate into broader public appeal. The role of political agitator is not the same as trade union leader, and if Crow really wants to radicalise the public, it's no good endlessly blaming the media for his unpopularity, as he does, when he knows what they are like. Has he ever considered moderating his profile, in the interests of bigger political goals? "Well what do they expect me to do," he protests. "Instead of going to Millwall, play croquet? Or go have a game of polo? At the end of the day, you are what you are."

But sometimes what he is is just not very believable. Inevitably, perhaps, he is much more likable than the tabloid caricature of an old union bully, and throughout the interview he is patient, cheerful and often quite charming. But when he says things such as, "I'm all in favour of co-operating with management," his voice goes noticeably higher, and when I ask him what sort of comments he gets from the public when his union goes on strike, it jumps even higher still: "I've got to say I don't get no aggravation."

Come off it, I say. That's just implausible. He'd be much more credible if he admitted that he sometimes gets abuse.

"I don't though," he insists. "I don't get nothing shouted at me. I use the train every day, I use the tube every day, I don't get people come up to me going: 'Oi, what about this strike?'" Doesn't he find that surprising? "Well when I walk along the platforms, I see people look at me and as I go past I hear them go: 'You know who that is – that's him.' I get that. But I don't get no aggro."

Does he think people might be scared of him? "Noawhh!" he exclaims, looking frankly delighted. "What would they be scared of me for?"