Let's do a little guided listening. I caught this on Portland's excellent jazz station, KMHD— If you don't have a good station locally, and you probably don't, you should be streaming KMHD live 24/7. The tune is Bags' Barney Blues, played by French saxophonist Barney Wilen, with Milt Jackson (on piano), Percy Heath, and Kenny Clarke.
This recording should tell you some things about playing the ride cymbal. Like, the interpretation you use does not have to be consistent all the way through the tune. Clarke plays with a strong quarter note pulse throughout— not with the accent on 2 and 4 many associate with bebop. You can also hear that he varies how he plays the “skip” note— sometimes he plays it with a triplet feel, with all the notes at a roughly even volume, sometimes he plays a dotted-8th/16th feel, with a very soft skip note. His comping is mostly played with a triplet feel, and the cymbal rhythm agrees with that— he's not playing triplets with his left hand with the dotted-8th/16th rhythm with his right.
Whatever he was doing leading up to it, at phrase endings Clarke often plays the triplet feel very emphatically, while dragging a bit, clearly conducting the time. Listen to his beat 4s. He's really bringing up the rear timewise here— he's way on the back of the beat, while Wilen plays very much on the front of the beat. Wherever Clarke is trying to take it, he succeeds, and the tune ends a little slower than it started. In his smooth way, he's playing assertively, acting as conductor of this tune.
With his comping, you can hear a lot of what I once called “the Kenny note”, which is actually a regular punctuation in swing drumming: a snare drum punctuation on the & of 3 in the second or fourth measure. As pure independence, it's an easy place to start when you're inventing a vocabulary for comping on the snare drum. We also heard a lot of that from Max Roach recently. In general, most of his comping happens at the end of the measure, and end of the phrase.