Mine (also 5 weeks) does the same thing, but we know she has reflux and are going to the doctor Monday to see about reflux medicine. It's awful to put her down, many times she'll spit up more than half an hour after eating (we try to keep her upright after feeding) and other times she'll just arch her back, make sour faces, and wake up screaming. Most times she'll be rooting for food even if it's been a short amount of time since she took down a decent bottle.
Someone suggested that our daughter may be trying to feed in order to soothe her throat. The milk coats her throat and helps ease the pain -- however, more food and especially more than she actually needs just causes more upset later. What we try to do is offer just 3oz, then use the pacifier (supposedly helps produce saliva to neutralize acid) and soothe her. If she's absolutely not having it and still giving strong hunger cues, then we give her more.
Try talking to her pediatrician about reflux; even if she's not spitting up she could still be getting it in her throat. Also, we were advised to feed in as upright of a position as we can, then keep her upright for ~30 minutes after if possible. The pediatrician also recommended we incline her mattress slightly BUT I would definitely talk to yours first before doing that; a flat surface is recommended for sleep UNLESS otherwise recommended by a professional. These things kind of help, which is better than not helping at all I suppose. I am hoping getting on medicine will help ease her pain.