Ascension levels are unlocked after completing a basic run through Act 3 with any character.
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.
This achievement unlocks after completing an Ascension 10 run with any character. The modifiers along the way are:
1) Elites spawn more often
2) Normal enemies are deadlier (deal more damage)
3) Elites are deadlier (deal more damage)
4) Bosses are deadlier (deal more damage)
5) Heal less after Boss battles (75% of missing health)
6) Start each run damaged (-10% health)
7) Normal enemies are tougher (have more health)
8) Elites are tougher (have more health)
9) Bosses are tougher (have more health)
10) Start each battle with an Ascender's Bane
in your deck
You'll want to commit to a single character as you work through the Ascension levels since progress isn't shared. I used the Ironclad to get through Ascension 20 but all three characters are viable.
Note that you only have to complete Act 3 in each run so you don't have to collect the three parts of the key to Act 4 or worry about beating the Heart.
Always plan your path through the acts. I found it best to hit as many ? rooms as I could while avoiding elites whenever possible. Remove Strikes and Blocks whenever you can to thin your deck.
I liked taking the starting blessing that caused enemies in the first three battles to have 1 HP, often allowing me to get to an elite for an easy victory. But if you take a different starting relic, Snecko Eye can be a viable strategy where you add more expensive cards to your deck and often can play them for 0 or 1 energy.
I found that getting a good +1 energy relic after the Act 1 boss made for an extreme difference in completing each run. Several have acceptable negative affects but I found Runic Dome and Fusion Hammer are awful.
Success with these runs will come with practice and persistence. Good luck.