Guide taken from planting42 on TrueAchievements, just being honest:
Each of these skills increase primarily after successful fights against Boxmore bots. You can also increase them by completing some character specific tasks and from the claw machine.
Level 5 is the max. I had most stats partway through Level 4 by the time I completed the story. I typically destroyed all bouncing boxes I found (bar the red ones as I found those to take too much time) with only buying occasional stat boosters. The stat I found to lag behind the most was Agility (middle stat - green sneaker) so if you find yourself in a similar situation - concentrate on completing your card collection then start buying those stat boosters when you can.
A note on boosters - they are all 1 use items. If it says x2 it means it will double the amount of that stat gained. If it says x3, it triples it. You can have a stat booster on all 3 stats at once if you wish though you probably won't need to work on all 3.
The game goes into an 'overtime mode' of sorts when you complete the story - the daily cycle continues and you can still earn 6 Technos a day clearing the plaza of bouncing boxes. Worth noting - you don't have to win the fights. Win or lose, the box disappears and progress is made. Winning is preferable as it increases your stats. This is when you'd be concentrating on using boosters to finish off the last bits you are missing.