In D1, Diablo wanted a host, so he corrupted Lazarus, who corrupted Leoric. Leoric was too old or in some way insufficient to host Diablo, so D took prince Albrecht instead. When you confront Lazarus, he thinks you're there on a hero's errand and taunts you because "you are too late to save the child."
After slaying D, the final cutscene shows the hero prying the soulstone out of D's head, which reverts the body to Albrecht's. Canonically, the warrior (retconned into Leoric's other son, Aidan The Insufferably Generic) wins, jams the soulstone into his own head to complete the prison, and becomes the dark wanderer. D2 happens. (This is also ~80% of the entire plot of D1. There was also a mushroom and some goats or something)
As for rising, Tyrael says all the undead are rising because he lost his justice juice.. or.. something. I just assumed that was it, Leoric wants a big ole' helping of justice for being used as a puppet, but is so insane at this point that it's kinda hopeless.
I think the overarching plot wasn't inherently bad, but the minutae and especially the presentation were horrible. The side stuff is well characterized, but easy to skip. The primary plot that gets shoved in your face with butterfly magic, VERY insistent warnings about upcoming boss fights, or just straight up telling you how to win a fight while you're in it (kill the shadow adds in Diablo's phase 2) is what sticks with players. It was also horribly atrophied on the details side as well, which would have been for the best if everything else wasn't so heavy-handed.
There were also
multiple butcher-type demons, the first one you see just has a nickname.