Here is the full list of all 12 The King of Fighters '98 Ultimate Match Final Edition achievements.
Achieve 50 wins in online fight.
Achieve 5 straight wins in online fight.
Achieve 10 wins in online fight.
Finish the game ( can't lose one round, can't continue) ( Arcade/Single Play Only ).
Highest point exceeds 300 thousand ( Arcade/Single Play Only ).
Achieve straight win ( Arcade Play Only ).
Finish the game (can continue) ( Arcade/Single Play Only ).
Pass level perfectly in one round (without injury) ( Arcade/Single Play Only ).
Iori(Ura) or Leona(Ura) appear as CPU ( Arcade/Single Play Only ).
Achieve 10 straight wins in online fight.
Achieve 100 wins in online fight.
Perform an assist attack ( Arcade Play Only ).