Switch outlets to permanently hot. Switch to control new recessed lights
You are in luck for making the outlet always hot. It looks to be wired as a simple switch loop. 

The only change required to make the outlet always hot is to separate the existing wirenut and replace the black on the outlet with the black that was pigtailed.

You hadn't given any details of how you expect the switch location to control the new lights but if it is wireless or you will fish a new cable to the switch you can pigtail white to white and black to black in the outlet location to get a neutral and live to the switch location for power from that circuit. Make sure to remove the electrical tape marker on the white that marked it as a live.