You unlock Ascension mode after completing a basic run through Act 3 with any character,
From the Slay the Spire Wiki:
Ascension is a game mode that adds new challenge modifiers to runs for added difficulty. https://slay-the-spire.fandom.com/wiki/Slay_the_Spire_Wiki
In order to unlock Ascension 1 for one character, you need to beat all three acts with that character.
Winning a run in each Ascension will unlock the next Ascension for that character only.
Losing a run will NOT reset any progress or decrease your Ascension number. Losing a run in Act 4 will still unlock the next Ascension.
You may play any Ascension level you have unlocked.
All Ascension levels can be played in Custom Mode, but completing them there will not count for your progress in the game.
The negative effects from each Ascension level are cumulative. For example: If you are on Ascension 3, the disadvantages of Ascension 1 and 2 will be in effect as well.