There is no level cap.

There is a level exp cap, that scales until you hit level 20:

As of a recent update, every level past level 20 now only requires 20,000 exp to reach the next rank, and this is consistent even after "prestiging". So once you hit Level 100, then reset back to level 1, but with a new prestige border icon, you will still need 20k exp to get to the next level, and so on.

New portrait borders are given every 10 levels, starting at 1, 11, 21, 31, etc.

After you hit level 100, you "prestige" and reset your level to 1, but gain a new portrait border. This happens every time you hit level 100 again.

At level 1890, or approximately your 18th "prestige" and at level 90, you will finally gain the last portrait border which appears gold and although you can continue prestiging over and over again your profile border will no longer change.

From an interview with Michael Chu, a Senior Game Designer on Overwatch (emphasis mine):

If you look at a game like World Of Warcraft, the Horde/Alliance split is such a big deal and such a defining aspect of the game. But Overwatch, pretty much from the start, we didn’t want to restrict ourselves that way. Because the game is so much about having the flexibility to play all these heroes and finding all the ways they play well together or against each other. Because of that we decided that the gameplay of the Overwatch universe wouldn’t be bound by that.

Our story justification – what I usually tell people – is that I think of it as, it’s clearly not canonical, but I think of it as how in comic book stories sometimes Iron Man and Captain America fight each other. Maybe in this instance on this map they’re fighting – for some reason they have some ideological difference or something that’s bringing them against each other.

But I think ultimately it was important for the game that you were able to play whatever characters you want. Blizzard has this development philosophy which is ‘Gameplay first’ and this is definitely an example of where we’ve gone with that.