You can earn these upgrades at ANY time just by going into a barbell-marked training facility and turning in a badge. You earn badges by doing random missions or regular missions. Note that you lose badge points when you get killed or do bad actions.

Upgrade 1 - running attacks
You must run and hit attack button on dummies. Hit 12 of them to earn the skill. I got 22 so it's pretty easy.

Upgrade 2 - Back Attacks
This involves attacking an enemy behind you. You have to get 12 of them. The trick is to spin past the enemy so it stays behind you and then use the attack key.

Fighting Training Areas
2nd Street
Wall St & 12th
Hoover and Francis avenue in koreatown

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