I have been at it since July trying to get a domination victory on Deity, with no luck so far.
I'm certainly willing to help you, but there are a lot of resources on these forums, just so you know.
I have used the Mongols for all of the games
Why? They're a good civ for Domination, sure, but there are better, and as others said above, they are not the best for learning how to Dominate. Try Poland/Babylon/Persia/Arabia/England as well.
I put my second city in a good spot, but apparently too close to a psycho civ and they DOW me before I have a chance to build good defenses
What is the build order in these games? If you think you might be irritating a civ by forward settling, send 3 archers and a Spearmen to the city, preferably with promotions from fighting barbs, and go to Construction ASAP for added defense.
In many cases, you might not only be able to repel the attack, but also conquer. For instance, Attila and Ashurbanipal spam their UUs and since they do no damage to units, they can be picked off near your city and a clear path is made to their own.
If I get past the early game, I can pretty consistently get Keshiks by turn 110 or so.
Chivalry by T110 isn't terrible, but try to do it faster. Don't get any unnecessary techs and just beeline it. Grow your cities so they produce maximum science and have a LOT of Chariot Archers ready to be upgraded.
I can usually get one capital finished off by Keshiks before turn 150
I think THIS is the crux of the problem. That is very slow going. You should have 2 or 3 capitals done by T150 as Keshiks are running out of usefulness soon after on Deity. With any other civ, you'd be relying mainly on XBs and Knights, and looking to hit Canons ASAP and then Artillery. Sounds like you are getting bogged down in the city capture part of things. For one thing, pick the shortest route to an AI capital. And if you have to capture cities along the way, raze them unless they have a Wonder you need or are just an excellent city or unless you have to keep it for strategic reasons. You want to hover outside of the city, baiting their units with workers, until the units are not threatening your XBs/Keshiks, then move in and capture the city in 2-3 turns. If the untis you have can't take the city in 2-3 turns, you are behind in the tech curve or taking on an enemy that has excelled along the tech tree and has defenses that are too good.
don't have a lot of friends if I took two capitals
Are you demonising your victims properly? Whenever possible, bribe them to DOW the entire world on the turn before you aim to attack. They will, and they will even sometimes send their troops away to fight in those wars, if they actually like the idea. Then Denounce, then DOW. Then the warmonger penalties will be lower and if you've nurtured a couple of good friendships and always paid your victims to attack those friends, then you can keep them all game.
It's that murky middle period of the game that I haven't quite figured out yet. It's that period when everyone hates you, so it's hard to trade for anything, you are behind in tech because you focused on beelining a military unit, and the front you have to defend from all of those AIs that hate you is much bigger now that you own a couple of their capitals.
Most of the UUs are coming in the Classical or Medieval eras, which still gives you a decent amount of time to focus your tech towards Education, and then either Artillery or Flight. So what I'm saying is that once you've built two armies, your cities should really be grown and nurtured so that you won't be as behind in tech. See also previous comments about doing diplomacy right.
May I suggest that you join in with the Community Challenge series I've just started? We're playing Immortal and Deity maps, interesting ones pre-selected by me, and we're sharing our playthroughs. It might help you learn?