In an announcement today, Blizzard said that players will "Earn experience while playing Blizzard matchmaking modes and custom games on Blizzard maps. You’ll earn XP for the race you’re playing with [and] each unit you build or destroy." Your experience is totaled at the end of every match and there are twenty levels for you to climb your way through for each race.
Levelling up lets you earn decals or portraits, and "These rewards get more and more epic as you progress in a race." I'm not quite sure how it is that a decal can be epic, but I suppose things have moved on from my time. When I was young, this was all fields and in my day we used to spell "levelled" with two ls, so I ain't changin'.
Blizzard say that the system will be introduced as part of the next beta patch, with testers able to start earning XP in the beta.