In one of the Classic Who stories the Doctor is called up for a crime he doesn't commit, they all wear funny outfits and hate him for being a renegade.
In one of the EU stories the Doctor's brother Braxiatel (one of the best villains in the entirety of scifi, in my opinion) talks to other versions of himself to lay his hands upon certain items for his Collection but it's never explicitly stated whether these are alternate selves or different versions of him in the same timeline.
In one of the Eighth Doctor novels Griffin the Unnaturalist is collecting different creatures and the Doctor and his companion end up in his collection.
There's also a massive class divide on Gallifrey with the ruling elite Time Lords having access to regenerations whilst the lower class ones and the shabogans don't, just like Rick and Morty has a class divide.
The Doctor also has to pull all the Time Lord asses out of the fire every time something bad happens, just like C137 but C137 is a bit more openly homicidal (the Doctor has a higher bodycount than C137 though.)
The Time Lords also have operatives who conduct interventions called the Celestial Intervention Agency who do morally questionable shit and meddle.
If you listen to some of the audio adventures, especially the ones written by Robert Shearman (especially the Chimes of Midnight) they have a lot more black humour like Rick and Morty.
I think it doesn't show up much in the non Citadel episodes, but Rick behaves a lot like the Doctor. And y'know the Doctor is a grandfather too who used to travel with his grandaughter Susan. (Who he eventually abandoned)