About a month before he died, my uncle had his back covered in lard...

About a month before he died, my uncle had his back covered in lard...
After that, he went downhill fast.

This is another pun. To "go downhill fast" could be taken literally or figuratively. The  literal meaning is to travel down a hill quickly, and the figurative meaning is for something (his uncle's health in this case) to decline rapidly.

The image of someone sliding down a hill on their lard-covered back is sort of a funny image. That is probably a part of this joke.

It might be funnier if the joke provided an actual mechanism for his uncle to have died from the lard - for instance if you changed "a day" to "an hour", then you could imagine the injuries from the trip down the hill to have caused his death.

Although, come to think of it that would probably also change the category of this joke.