Requests pop up in tour mode sets (You must be playing the TOUR MODE, this is the main career mode), normally replacing the final song of a set. You'll have the option of playing either a new song or the one you selected.
To 'nail' the request you have to get 5 stars on the song. I'd suggest picking an easier song if going for this achievement but usually requests come from the same difficulty level as the set you're playing. The achievement should pop when the song ends.
Note that there is an achievement for Gold Star-ing a request (All players must be on expert, usually Gold Star cut off is about 1.5 times the 5 star score) so you may want to pair them together.