Did the monkees play their own instruments?

Yes and no. Mike Nesmith was a proficient guitarist before joining the Monkees and Peter Tork was a fairly proficient guitarist / keyboardist. Micky Dolenz was a rudimentary guitarist but had no experience in drumming before joining the group and had to have crash course lessons in drumming before the group went on their first tours in 1966/67. Davy Jones didn't play any instruments other than the tambourine(!) although in the film Head he can be seen and heard to play organ on the live version of Circle Sky.

Most of the criticism about the boys not playing their own instruments was disingenuous at best, often perpetrated by artists of greater skill but lesser success, or by critics who compensated for their own lack of musical talent with insufferable arrogance. Most bands at least use additional musicians to enhance their performances when recording in a studio. Rarely do artists provide all of the instrumental or vocal performances on their own albums; some actually provide very little or none of the instrumental performances or background vocals on their albums--a practice which so widespread during the 1960's and the 1970's that it could truly be said that very few artists during those periods actually performed anything more than the primary vocals on their albums. And although the boys may not have all been accomplished musicians when the group was originally formed, they quickly became at least competent musicians as evidenced by the fact that--after they managed to get Don Kirshner canned--they were finally allowed to perform on their own albums and in their live shows.