I’ve found a ’85 535i that I’m considering buying as a daily driver for my son. Seems solid and has about 200k on the car. Asking price is cheap at $1500 and has some cosmetic issues. My primary concern is this. When shifting the car from 1st to 2nd, I feel a vibration coming from the rear of the car. Vibration is from 15-30 mph. Changed the right rear tire and still has the same vibration, so not a tire issue. There is a broken outside boot on passenger side rear axle and just starting to sling grease. I don’t want this to turn into a $3000 car with repairs. What are the possible issues associated with the car?

TIA

Probably the center support bearing, or the flex disk (guibo). You can fix both for about $100 in parts. Obviously you'll want to repair the driveshaft boot too.
As long as the car isn't horribly neglected it should be pretty easy to make a solid daily driver. It wouldn't be a bad idea to take it to a trusted BMW mechanic/guru to get it thoroughly inspected.
But in general, e28's are pretty cheap to keep on the road.
Good luck!