what does he actually do?
Actually? Nothing.
Technically, according to our Constitution, he's not even in the line of succession for the presidency, if a big icicle falls on Putin's head one day from the Kremlin bell tower.
He's Deputy Chairman of the Security Council of the RF. The Security Council is less powerful than the American National Security Council. It's a purely consultative organ, with no executive functions.
As Deputy Chairman, he does not even control the agenda of the Security Council. Putin sets the agenda, and chairs the meetings.
He's not even the second, or third most powerful person on the Security Council - that's the Minister of Defense and the head of the FSB.
--//--
So he doesn't do much of anything.
That's why in recent years, Medvedev has largely reinvented himself as a radical hawk, posting unhinged threats on Telegram and echoing the most maximalist war rhetoric. But he does it in such comedic fashion, that even his supposed constituency of "war fighters" can't take him seriously.
He wears a Mao-style French tunic, and strange jewelry. Likes to announce impromptu visits to state armament factories, and then prowls around the production floor nodding at tanks and artillery pieces with a knowing wink and a little pat of his paws.
They say he writes too, but the things he writes are so hideous - that people need to check themselves into rehabilitation for two weeks after reading his works. Allegedly it's poetry, but I have also heard fantasy fiction.
Curious fella.