Good on you for continuing b/feeding. As you obviously already know, it is by far the best thing you could be feeding your baby. However, it doesn't mean that it is great for your own well being sleep-wise. Every 3 hours'ish sounds about right for a 5 week old. It is all a bit of a blur for me now, but from memory I don't think my daughter slept longer until about 3 months, although I did get the occassional 5 hour break.
If you are really finding it difficult, perhaps you could express some milk before going to bed and have daddy feed for at least the first wake up? I have to admit this never worked for me because 1) I was hopeless at expressing, 2) my daughter would never take a bottle and 3) if she was awake, I couldn't sleep anyway, so I figured I may as well get up and feed her.
Although it doesn't feel like it today, babies are only so tiny for such a short period of time. You'll look back and be so glad you stuck with b/feeding.
Best of luck.