You can see Captain America, as well as a lot of other Marvel heroes, in action now during Captain America: Civil War, and Chris Evans will reprise Steve Rogers in the upcoming Avengers: Infinity War movies.
Of course, because of Captain America: Civil War (no spoilers here), Steve Rogers has gone rogue in protest of the Sokovia Accords and superheroes being regulated by the worlds governments.
Just like in the comics, Captain America was considered dead following his last mission in World War II.
Near the end of World War II in the Marvel Cinematic Universe, Captain America crashed Red Skulls plane full of weapons of mass destruction into the Arctic.