This collection of trophies can be time consuming (between 20-30 minutes per ladder). Thankfully it can be completed on Easy difficulty. Just select PaRappa and run through the 8 battles (detailed below)...
1 opponent
2 opponents - FFA
3 opponents - FFA
3 opponents - FFA (Harder A.I.)
3 opponents - FFA, First to 3 kills
3 opponents - FFA, First to 3 kills (Harder A.I.)
Rival Battle
Final Boss Battle