MUMBAI—One of India's most famous police officers is on trial—accused of being a killer-for-hire—in a case that embodies the difficulty of trying to clean up the nation's notoriously corrupt crime-fighting forces.

The officer, Senior Inspector Pradeep Sharma, was once so widely revered that two Bollywood movies have been inspired by his exploits. They include the 2004 hit "56 So Far," a reference to the large number of gangsters killed by the big-screen hero.

Today...