My oldest didn't start sleeping through the night until he was almost 5-months old. My second started at 6-weeks, but at 5-months started waking up again and didn't sleep through the night again until 10-months. With my son, it seemed somewhere between 6 and 8-weeks he really started to fall into a pattern of waking up only to eat for 10-15 minutes and then right back to sleep.
It is really hard to deal with the lack of sleep, but after speaking to many, many moms, it seems that sometime in the second month things just magically tend to fall into place. The baby is sleeping better (not necessarily through the night), your body has adjusted to the new sleep schedule. You are just more in-tune with your baby.
For now, sleep when he sleeps (dishes, laundry, etc can wait). If possible, see if your husband could take a feeding or two at night so you can get at least 4-6 hrs of uninterupted sleep (it will do wonders for you!). And I admit that there were a few nights that I was so desparate for sleep, I brought my kids to bed with me. It wasn't the most restful sleep, but better than none at all.
If you're not already doing it, keep things really low key at night - no playing, talking, bright lights, etc. Keep it really just to the business of diaper changes and feedings. During the day, play with him a lot, keep the blinds/curtains open. Keep noise levels normal. Teach him that night is for quiet and sleep, day is for fun and being awake.
Hang in there. It will get easier!