"war crime is when war I don't like and the more I don't like it the war crime-y-er it is" - average person discussing war crimes of modern presidents
For the record, Bush is guilty of legitimate war crimes (sanctioning the torture of PoWs) and crimes against peace (the unprovoked and unjustified invasion of Iraq), and Trump is quite possibly guilty of war crimes (his advocacy of 'killing the families' of enemy combatants combined with the fact that he removed the transparency from our drone strike program is... suspicious, to say the least) but shit will leak out from that administration for years to come yet.
Obama and Biden probably do not count as war criminals. Killing enemy combatants is legally unproblematic. However, the acceptance of high collateral damage and relatively low certainty of identification may qualify them, depending on how you see the matter of drone strikes vs. traditional methods. It would be a much tougher sell at the Hague.
EDIT: It was pointed out below that Obama is more likely a war criminal due to the approval of legally dubious 'double-tap' drone strikes which presume first responders to militants are militants themselves.