To start the battle, wait until you're transported to a large arena. The Archon has two basic attacks and the one in which he sends a cluster of stones at you is by far the most difficult to evade. Try to figure out when to dodge exactly, and each time jump to the side, not backwards.
In the other attack, the Archon shoots out a black grease towards you, but this is something you shouldn't have any problems dodging. Apart from these two types of moves, be ready to dodge the Archon's standard melee attacks, but if you're busy hitting him yourself, you shouldn't witness them too often.
While fighting the Archon, I suggest applying the well-tried method of pulling yourself to him after each of his (and missed) attack. To do it, use Death Grip of course, and then perform a series of quick hits.
Keep hitting the Archon until he loses enough health points to prompt him to start flying over the arena (the above screen), rendering himself invulnerable to your standard attacks.
This is another stage of the battle in which you need to keep an eye on two different types of the Archon's moves, which are: a pound on the ground (screenshot 1) and throwing boulders (screenshot 2). Keep a longer distances than before - it will let you dodge easier.
As I mentioned before, the Archon is invulnerable to standard melee attacks while he's in the air. Make sure you have Dead Grip selected and wait until a well-known purple icon appears on the his body (screenshot 1). Pull yourself to him (screenshot 2) and hit him with what you've got.
Repeat the process (pulling yourself to the flying Archon when the opportunity strikes) until the boss falls to the ground. Run to him and hit him until an icon indicating the possibility of performing an Execution appears (the above screen). Press E quickly to trigger a cutscene and you'll proceed to the next stage of the battle.
This new stage is very much similar to the first one, the only difference being that the boss will only attack you more successfully with boulders. Consequently, pay more attention to dodging. Keep attacking with Death Grip and repeating the same process until the Archon loses enough energy to make use of his wings again.
The last stage of the battle is only as difficult as difficult it will be for you to dodge the Archon's charges (the above screen) - which shouldn't be too much of an issue. Keep using Death Grip to pull yourself to him every time he misses.
Continue the battle until the boss loses almost all of his health and wait for the Execution icon to appear again. Pressing E at the right moment cements your victory, rewarding you with the Angel Key.