I still need to find out a good place to get the entire lore, honestly.
An Overwatch wikia will probably do the trick. Lore has built up over the years, but there's not really
that much lore.
'Overwatch' used to be a team of heroes that were united from around the globe to go on missions like protecting a dignitary or tracking a terrorist group etc.
At some point in the past before the game, (bits and pieces of what 'truly happened' are teased more each year) something happened to cause Governments to enact an official disbanding of the Overwatch group (I guess you can think of them as having been S.H.I.E.L.D. and the main terrorist group they fought, T.A.L.O.N., was like Hydra).
The game's 'present day' (as implied by watching the opening) is Winston trying to get the team back together since he knows T.A.L.O.N. is still active and the world needs them; whether they realize it or not.
New Characters have appeared (roughly 4 a year; dlc of characters, maps, and events have always been free) that add to the lore and 'progress' the story little by little. I'm not super positive on the timeline, but I believe that Orisa (an AI centaur robot created by a 13yr old girl in Somalia I think?) is the 'newest' member. Game release wise, Ashe was the last (female western-themed girl with a rifle; think Bonnie like Bonnie & Clyde) and Sigma (a Dutch Astrophyscist) will be releasing soon (he's currently on the PTR; Public Test Realm; so bugs can be worked out and players can play test him).
EDIT: Nevermind; Baptiste was the most recent release; I forgot! He's a healer from Haiti)