I just can’t seem to get the hang of doing a purl stich while knitting with double-points in a circle. I am now trying to make this hat with 2/2 ribbing, so will I k2p2 all the way around and when I start the next round, do I p2k2 since I am going in a circle?
Yes, you K2P2 all the way around for your ribbing.
When you get back to the beginning of your round, you also K2P2 around again. The knits will be the smooth VV of your rib, and the purls will be the bumpy part which will appear to recede into the tube.
Always knit over the knits, and purl over the purls for circular ribbing.