It's getting there. I saw a video last week of someone running Overwatch in wine at about 45 FPS with a patched version of Wine. Not sure how easy it is to get running or if/when those patches will be merged in, but it's making good progress.

From what I remember, Overwatch doesn't use much of DX10/11, possibly just a few minor things, so it's mostly just a case of getting it not to crash when those features don't exist, and the rest should perform ok.