Character bios
A great place to start, it will give you a solid base to stand on going further. Recommended reading order:
Reinhardt
Torbjörn
Mercy
Winston
Tracer
McCree
Pharah
Widowmaker
Soldier: 76
Reaper
Mei
Bastion
Zenyatta
Hanzo
Genji
Roadhog
Junkrat
Zarya
D.Va
Symmetra
Lúcio
In-world news stories
Fading glory: On the trail of Jack Morrison (The rise and fall of Overwatch)
Experimental weapon stolen from Watchpoint: Grand Mesa
Mei’s adventures
Lúcio brings music to life with Syneasthesia World Tour
A Moment in Crime Special Report: “The Junkers”
Origin stories
Comics
Train Hopper (McCree)
Dragon Slayer (Reinhardt)
Going Legit (Junkrat and Roadhog)
Animations
Recall
Alive
Are You With Us?
Cinematic Trailer
We are Overwatch
Launch Celebration Trailer
Congratulations, you should now be up to date with Overwatch lore.
Lorewatch (the playlist is backwards, so you have to start from the bottom)
Overwatch wiki
Google spreadsheet
Bonus:
Heroes of the Storm - Tracer Trailer
(Last updated: 2016-05-11)
Have I missed a source? Please let me know.