How can I check the rank of a player after a match?

Under your profile in the game(as well as outside the game at the Starcraft 2 website when you login to your account), there is a section called Match History where you can see all your previously played matches. This includes games from previous seasons. You can filter on 1v1, 2v2, 3v3, 4v4, FFA, Cooperative, and Custom game matches.

Unfortunately, this does not show you the name of your opponent until you click into the game in the Match History selection screen, so you'll have to remember the approximate time, game type and map on which you played the game to find previous matches.

Once you find the game, you can click through to find your opponent's profile and their rank. Unfortunately, you can only access the match score screen from within the game, not on the website, so you will need to boot up Starcraft 2 beforehand.