Hi S.--First of all, you're doing great. He'll be fine, you'll be fine, this is just the roughest part to get thru.
At 5 weeks old, your baby will be asleep more than he's awake. But, at night time, he really isn't capable of sleeping longer than 4 hours max at a time--no matter what you feed him before bed. (Cereal is not advised till 6 months, except for babies with reflux.)
He's going to wake up at night, maybe a several times, for awhile longer because he needs to eat. When you get up to feed him, try this approach that worked for me. Keep the lights and noise off. If you're formula feeding and baby likes warm milk, put the bottle in a warmer, then go get baby. Be quiet and gentle, change the diaper if needed. Feed baby, burp baby, rock baby back to sleep. If baby just falls right to sleep while eating, I wouldn't burp unless you know you can do it w/o waking him up. Check the dipe again, if you can w/o waking baby. (This is why I always changed before feeding--feeding usually puts baby back to sleep, and I didn't want to wake her back up by messing with her diaper. You can also use the mom-trick of just smelling his bum, gross I know, to determine if he needs another clean dipe.)
I know you must feel like you're losing your mind with sleep deprivation, but it does get better. It helps a lot if every 3rd or 4th night, someone can take the night shift for you, so you can sleep straight thru. Each week, he will get better and start to sleep longer at night. He still needs naps during the day, lots of them! It's not true that keeping a baby up all day will make them sleep all night.
Also, ask your pediatrician to tell you about common growth spurt times--I think they are at 6 weeks, at 2 months, and others I don't remember, in which babies commonly get really really hungry frequently for a few days to a week, to stock up calories before they grow a whole bunch at once. It's not a foolproof plan, but maybe you could arrange for extra help around the weeks of growth spurts? I don't know what other moms would say about that. My daughter definitely followed the growth spurt pattern.
Good luck! You're doing great! I think most of us would agree that after 8 weeks, and then again after 12 weeks, things felt much easier.
N.