Due to requirements most will require New Game plus.

Achievement	How to unlock
Beyond Time	Defeat Lavos after re-acquiring Crono.
Dino Age	Defeat Lavos after battling Magus but before battling Azala.
Dream's Epilogue	Defeat Lavos in the Ocean Palace, finish the Dimensional Vortices, talk to Gaspar, use the bucket Gate to Time's Eclipse, defeat Dream Devourer.
Good Night	Use right telepod, or use bucket at End of Time, defeat Lavos before leaving End of Time.
Legendary Hero	Defeat Lavos before you get the Hero Badge, but after you hear Tata is the hero.
Memory Lane	Defeat Lavos after Schala uses her pendant, but before Marle's pendant is powered.
People of the Times	Defeat Lavos after getting the Gate Key back but before repairing the Masamune.
Reunion	defeat Lavos after Crono's death but before re-acquiring him.
The Dream Project	Use right telepod, defeat Lavos at the very beginning OR defeat Lavos at the Ocean Palace.
The Oath	Defeat Lavos After giving the Masamune to Frog, but before facing Magus.
The Successor of Guardia	Use right telepod, defeat Lavos right after returning from 600AD.
The Unknown Past	Defeat Lavos after receiving the Hero's Medal from Tata but before the Gate Key is stolen in prehistory.
What the Prophet Seeks	Defeat Lavos after defeating Azala, but before going to the Ocean Palace.