I will say this from the start: this method of farming Azmodan is not my own discovery. I saw it a couple of days ago in a video that showed how to level low level friends with your high level character. Unfortunately I can't find the link to that video.
Apart from explaining how to do this (if you haven't seen the video and don't know what this is about), I will also put some numbers to the strategy, namely how much XP and gold you can expect for each farming run.
Just keep in mind that these Azmodan runs are primarily for experience, not gold or loot. The boss himself doesn't drop anything of significance, unless you get lucky with a blue item with perfect stats. If you do a full clear of the zone leading up to him, you can expect to find a group or two of champions/rares, as well as loot in the form of gold and items of all sorts.
This strategy revolves around the quest called "Kill Azmodan". You should have access to this quest once you've finished Act 3. Because I have moved on to Hell, I can go back and farm him as often as I please. The beauty of this is that after killing Azmodan you will get the quest credit in the form of XP and gold and then you will get another identical credit shortly after when you return to the Keep.
1. Select the quest "Kill Azmodan" from the quest interface. I do this on Nightmare. It is certainly possible to do it on Normal or Hell if you wish. I'm guessing that on Hell it won't be very lucrative because you are probably already 60 by the time you will tackle this quest. But whatever.
2. Log into the game and teleport to the last waypoint on the list, called "The Core of Arreat". From there on, head in a northern direction until you reach Azmodan's lair. The layout of this zone never changes so you will never get lost.
You can choose to skip the monsters in this area, if you can get past them quickly without being killed. Stunning/fearing/disabling them will obviously be of great use here. If gaining the most XP in the shortest time possible is your only goal, then this might be advisable.
3. Do your thing and kill Azmodan. You will get the first quest reward.
4. Teleport back to the keep. Right next to the portal is a soldier called Lieutenant Lavail with a yellow quest mark over his head. Talk to him and he sends you to the Armory and from there immediately to the Keep Tower. Go there.
5. Once you get to the Keep Tower you will receive the second quest reward. Both rewards are identical and that's what makes this very lucrative: you get at least 89K XP for this quick run.
6. Log out of the game. Rinse and repeat.
To make this quick and safe you will obviously need a lot of DPS and health. To give you a reference, my level 51 Monk has 2200 DPS and 14K health and this run is a walk in the park for him in Nightmare. It shouldn't be too hard to acquire equipment that gives you these stats considering that there are others who are much better equipped than I am, at the same level.
Occasionally you will run into groups of champions/rares with some ugly affixes on them. It's not worth dying for these idiots. Just run back to the waypoint and teleport back to town where you can log out safely and restart the quest, or alternatively, try to run straight to Azmodan, bypassing all the nasties.
As you can see from the images above, each of the two quests gives 44,800 XP and 1940 gold.
A typical run netted a total of 127,628 XP + 7000-8000 gold from gold drops /
quest rewards / vendoring drops. Then there's a variable amount
from gems and crafting pages (AH value) that you will also find.
To get from level 52 to 53 you will need 10 such runs. Each run takes 5-7 minutes (with full clear), meaning that you can level your character in only 1 hour with Azmodan runs, on Nightmare difficulty.