Buddy Pine unlocks with the completion of Level 7 - Golden Years. No special action needs to be done to unlock him other than completing the level and buying his red character pack.
Syndrome unlocks upon completion of the last crime wave. There are 10 in total, each taking place in a different district around the town. Very straight forward completing the quests in each town and besting the villain who is attacking.
Once these two are unlocked, have both out in the main world and the achievement should pop.