This is pretty straight forward to do, but just takes a little time.
In the bottom right corner of the screen is a golden s shaped bar (on its side) that fills up as you gain Karma from killing enemies. Each time this fills up you will gain a Skill Point (which is 1 level). There are a few points to note :-
1. If you die while in the campaign, you will lose all Karma gained on your current level, however I found a work around. If you think you are about to die … jump off the map, that way it spawns you at your last checkpoint, Karma intact.
2. If you play the Purification Challenges (mainly Purification II) and you die, you keep your Karma, so this is a quick way to level up, once they become available.
3. There are also Skill Cards that give more Karma and some of the upgrades give loads. At one point I had +40% extra Karma per kill
Hope this helps