I can't remember when mine slept through...maybe 4months? Thats was with a dream feed at 10pm. Then she would be up at 6.30ish.
I dont think any baby sleeps through at 3 weeks but I would start trying to help bubs to work out night and day cause they dont seem to automatically know this. During the day make sure it's always bright and loud and play with baby when he is awake. At night feed baby in as dark a room as possible, dont talk to him when changing his nappy and try your best to put him back down.... this often wont work so maybe rock him in a dark room without speaking.
Hopefully he will work out day and night soon and go a few more hours without waking.
Also are you letting him feed til full or til he finishes his bottle. Babies are not very inclined to overeat at that age so if his finishing his bottle perhaps try putting some more in the bottle.
Not sure if any of that helps but I hope you get a rest soon. Maybe see if a friend can come and help you out so you can get a little un-interupted sleeep?