I have one more ending to experience to get the last of the 3 ending achievements. Do I need to experience it on the same character I did with the rest of my achievements? Or can I start a new character to experience the final ending and get the achievement?
No, the endings can be done on separate characters and you will still get the achievements. I personally got two of the endings on one character and the third on a separate one.
However, to get the achievements for collecting all spells/miracles/pyromancies/rings/gestures, you will need to collect them all with the same character.