I second the routine, and plenty of feeds, but I also think it is important to know that you can let your baby cry for five minutes.
Go through your routine, have the last feed somewhere dark and peaceful (not infront of the TV) then put baby down and walk away. If baby is still crying in five minutes then back in - reasure and cuddle, still in dark peaceful place, then back in bed. If you've done a good feed, then more food isn't necessary for a couple of hours.
Lots of people will say that 5 weeks old is too young to try this sort of "controlled crying" technique but I think it is important to get the message, "it is bedtime therefore you sleep" message in early.
I have 3 girls, oldest is now 10, who all slept through the night from 6 wo by giving them this strong message. My brother also has 3dc, none of whom slept the night because if the children were awake then they were played with/rocked what ever time of day or night it was.
It does get better, but if you are going to be good parents you need sleep.