My son's 14 weeks and he nursed every three hours like clockwork for probably the first 6 weeks-- more frequently during growth spurts. He didn't start going longer stretchs until maybe 8 weeks. Breastmilk is indeed easier to digest and most newborns need to nurse every two to three hours. 5 weeks is too young to really hold out much hope that you're going to be getting a lot of sleep at night. One thing to at least help her clear up days and nights is to not let her sleep longer than two hours at a stretch during the day (and if this means waking her up so she can eat, do it) and to make sure that at night, you are keeping things night-oriented: lights off, little talking, no play. And don't feel like, during the day, that you need to keep things quiet for her to nap.
Cheer up, the end to the every few hours is probably close. My son started to go longer stretchs at night by 8 weeks (it was inconsistant, but he could do it) and now he's down for bed at 7PM, and sleeps till 1 or 2 in the morning, then up again by 6. One thing to realize is that, for a small baby, the term "sleeping through the night" actually means 5-6 hours, not an exhausted adult's definition. My pediatrician did say that once my son showed us he could go longer stretchs, my husband should go to sooth him if he woke before that point but personally, we didn't do that. We've only just started that now, soothing him from 5AM on to hit the 6 mark for his morning nursing. The doc did specify the dad, as a nursing mom smells like food.
This part is just me, but you may want to think about nursing him at night (I know, it's exhausting) but for some women, the baby is way more efficient than the pump. It's possible she's waking frequently because she wants more milk than you can pump. Best of luck, it really does start getting better.