How can I play vs A.I. on a set difficulty on a random map?

No.

They changed the way match-making works, and it is no longer possible to select the difficulty of the AI you play against (other than perhaps going out of your way to find people who are currently up against the difficulty you want). Even switching back to Wings of Liberty mode will not let you use the old match-making.

You can make a custom map & open it to the public, or join a custom melee match, but this won't allow you to play on a random map, and it counts towards a different set of achievements.

You can still create a custom game with AI in them.

push the create game button, add some AI and go for it!

Or, you can simply do the training games (found in Matchmaking > Training) at your preferred level.

Not sure how custom games count towards achievements, though.