The ultimate question between PvP and PvE which is the best way depends on how long it takes to complete each respectively in Star Trek Online .Pvp works nice cause there's always extra missions you can work to accomplish ( always a mission that rewards a few 100 skill points, for completing x amount of games, in addition to kill missions), and its relatively fast. You would avoid capture and hold while leveling (map is fun but playing a single battle for 30 minutes plus for 150 skill points gets annoying),as well as ground pvp (again takes a long time). If you want to level up your character you can visit here and find the most secure sto credits.
PvE: again times the biggest factor, but every exploration completed out of the 3 rewards xp (like fed side but you dont get any crappy data gather missions). At captain 10/11 to BG5 you're getting around 6-800 skill points by the time the mission is completed.
Outside of which ones gonna level you the fastest, you should also consider that you'll make grips of money pvping (conservatively estimate that you made at least 4-5 mill over the course of leveling), pve missions will be the same variation of kill space/ground, scan ground, and pvp while the same maps is you playing against characters(sto credits) which can be more exciting.
Also (although you can accomplish this with explore badges, just takes longer) if you level through pvping by the time you're mid level of your current rank you'll have gotten enough pvp badges to upgrade everything on your ship.
The best way is which ever way appeals to you. From a specific drop/points/badges/...etc. Many players do exploration missions when they first crack rank... they prfer to go into PvP with the best equipment on their ship or person as possible. They find doing exploration missions takes very little time. If you've nothing to do for a week there is no reason you can't whip to BG in a week time. Just go with your instincts. Usually 3 triplet explorations per level. 30 missions later you've bumped to the next rank. Takes a few more as you go up in level, but still pretty fast.
"During the the campaign one group time and again reversed advance after advance by the Federation and its allies. All of the players of the Klingon 69th squadron have proven a terror to the allied players".