A new born baby has a stomach the size of a walnut. They aren't designed to go lots of hours without milk. Sleeping through the night isn't really healthy as young as 5 weeks, afaik, although it's more common in bottle fed babies (because cow's milk is so much harder to digest - they have quite different poo from breastfed babies)
If you're worried about getting enough sleep, then I'd recommend you do some research about co-sleeping and just take your baby to bed and sleep while you feed them. You won't care how many times a night they feed then! (Deborah Jackson's Three in a Bed is the classic book, there is also information at DRSEARS and several co-sleeping conversations going on in the "sleep" bit of this message board.
Most tiny babies don't really do "routine" to start with, tbh. Bottle or breast. All the advice now is that you feed them when they want it, and they sleep when they need to. The best advice I was given when I first became a mother was: "sleep whenever the baby does".
Yes, breastfeeding can work with a baby who sleeps peacefully. Most easily with co-sleeping, or baby in a moses basket right next to your bed so you can scoop them up, with pillows arranged so you can doze while they feed.
But please don't expect your baby to sleep undisturbed through the night when they are tiny. Some do, some don't, however they are fed. It's not to do with whether they are "good" or not, it's to do with how big their tummy is, and how much comfort and reassurance they need from you (remember, they've had your heart beat as a constant sound track for 9 months. Night time breastfeeding can be about continuing that reassurance as well - some people call the first 3 months the "fourth trimester")