Picasso's still lifes are less nature morte than life in extremis. He painted skulls and more skulls. His trussed roosters look more like people struggling on a torture table than lunch in the offing. Asked why he painted so many pictures of food, of pots and cutlery jangling in the drawer, of lamplight and gloom during the German occupation of Paris, Picasso declared: "A casserole can also scream! Everything can!" He was also hungry, as was most of Europe. But he refused extra fuel and food coupons, refused to collaborate.

The room of still lifes at Tate Liverpool's new exhibition is a killer arrangement of paintings and a small number of sculptures. An owl perched on a chair-back stares at us like an atavistic, simian head, in a painting as bare as an empty larder. A great bronze skull is a cratered cannonball, a damaged weight. One remembers that Picasso's last self-portrait, a drawing from 1973, the year of his death, is more skull than living head.

"A dead man in Spain is more alive than a dead man anywhere in the world," wrote the poet Lorca, who was murdered by Franco's henchmen, and whose bones are yet to be found and given a proper burial. The debate about the disinterment of the mass graves of the missing dead from the Spanish civil war is a major issue in Spain today – more than at any time since Franco's death in 1975. Picasso spent almost half his life in exile in France after the civil war, during which time the Republic made him director in absentia of the Prado museum. He refused to return to Spain while Franco was alive. And, because he was a member of the French Communist party, he was never allowed to visit America.

But despite Paris's decline during the 1950s and New York's ascendance as the centre of the art world, Picasso never much cared that he couldn't go there. Along with many other French intellectuals and artists, Picasso joined the Communist party in 1944 and remained a member for life. He channelled money into the party and into communist newspapers. He gave a million francs to striking miners. He made works, especially drawings, for communist-inspired peace conferences and innumerable other causes. He was, surprisingly – especially for a Spaniard of his generation – an anti-racist.

Picasso's politics were never in doubt, though this side of him is often pushed aside for a view of the artist as a protean genius and priapic monster. He was a man of his time, shaped by his upbringing, ambition and talent (it possessed him as much as he possessed it), as well as by the events he lived through. Guernica, commemorating the destruction of the ancient capital of the Basque homeland in 1937 by German and Italian bombers, and painted the same year, remains Picasso's best-known declaration of revulsion to fascism; but themes of war and suffering were a constant in his work. Guernica's blacks, whites and greys, as stark as newsreel footage and front-page news, were continued in works such as the less-than-successful 1951 Massacre in Korea, or the great Charnel House, now in the Museum of Modern Art in New York. The current exhibition opens with Charnel House, which would alone be worth a visit to Liverpool; but Picasso's small still lifes of the 1940s, 50s and 60s carry a similar symbolic weight, even if their meaning is more furtively delivered. They are filled with disquiet.

"Painting is not made to decorate houses," the artist wrote in 1943. "It is an instrument of offensive and defensive war against the enemy." Picasso toiled over Charnel House slowly, between 1944 and 1945, despite the fact that the painting appears almost cursory and unfinished. It is a deceptively complex and rich painting, with an amazing tension between the subject and the language used to depict it – the slaughtered family heaped dead under a kitchen table, their bodies intertwined. The more you stare at it, the more you get entwined, too. The painting was initially inspired by documentary footage showing the assassination of a family during the civil war; the ghost of Goya's Disasters of War hovers in its mangled stillness. This is nature morte as aftermath. An arm reaches upward, stiffened in death, the hand bloated and seamed like a baseball mitt, clutching at nothing.

As well as slaughters and still lifes, the exhibition is filled with posters, scarves, copies of telegrams from Fidel Castro and commendations from the Russian politburo. Curator Lynda Morris has spent years in the archives, gathering material. There are photographs of Picasso listening intently to speeches at a peace conference in Poland; Picasso with Soviet officials; Picasso staring at a photograph of Stalin.

All this is fascinating stuff, and details Picasso's commitment and generosity. He handed over suitcases of cash. He protested the death by electric chair of the Rosenbergs, executed for handing over US atomic bomb secrets to the Russians. And he made his only trip to the UK to attend a peace conference in Sheffield in 1950; upon arrival at Victoria station, he was detained by immigration officials for 12 hours.

But it is the art itself that really counts (though with Picasso, everything counts). He threw nothing away, and even the most minor drawing or note takes on a talismanic significance, as it did even when he was alive. Asked by the owner of a small bistro in the 1950s whether he would be so kind as to do a little drawing on a napkin as a souvenir, Picasso replied that he only wanted to pay his bill, not buy the restaurant. And although he was the world's most famous communist artist, and received the Order of Lenin, he refused to toe the party line and reinvent himself as a socialist realist. He remained a decadent formalist, as far as Soviet critics were concerned.

A horse with enormous balls

It is impossible, too, to say exactly how political some of Picasso's art was. Perhaps everything was, at some level – although I do find some of the claims made about his later work as tantalising as they are tendentious. The idea that Picasso's variations on Velázquez's Las Meninas might contain veiled references to Franco, and to Franco's grooming of Prince Juan Carlos in order to reinstate the monarchy, is unconvincing. (As it tuned out, the Spanish monarch later declared that he wanted to be king of a republic, and helped pull Spain back from a coup d'etat in 1979.) Were Picasso's variations on Delacroix's The Women of Algiers a reference to the Algerian war of independence? What they seem to be is a celebration of a cloistered, civilised bath-house, with their tiles and hookahs and disporting female flesh. The 1960s Rape of the Sabine Women, inspired by Poussin and David, may well have been a comment on American cold-war aggression and interventionism, particularly towards Hispanic America. A horse with enormous balls and grinning head stamps on a woman; a human foot kicks at a woman thrown from her bicycle. Picasso didn't need to modernise these paintings, adding rubber batons, riot shields or clouds of tear gas.

Picasso: Peace and Freedom covers its subject fitfully, and is dependent on what loans were available. Later in the show, we come to quieter images: Manet's Déjeuner sur l'Herbe redone cartoonishly, and very late paintings of musketeers, the artist imagining himself morphing into Frans Hals, and a horny old whiskered Rembrandt. These come as something of an aside. Asked about his political views in 1968, the artist remarked that if he wanted to respond to such questions he would change his profession and become a politician. "But this, of course, is impossible," he said. Art exists in the social world, and is political whether we want it to be or not. Picasso took the bull by the horns. His art stands up for his own individual creative freedom, but it's one that didn't give him much peace.