5 weeks is hit or miss-usually miss! He can DEFINITELY make it through the night by 3 months.
If you don't want to build a dependence on your comforting for sleep-smart move!-then don't use any sort of rocking and nursing to put him directly to sleep. Do all your rocking and cuddles during the day time and finally for the pre-bed routine, but once he's had and done everything, including his last nursing, put him cuddly in his bed, and walk away.
He may cry a little, but will probably fall asleep. Don't let it be too long at this age. Try to gage if it's fearful frantic cry, or just a normal, tired cry. A little normal crying is fine, he's tired and learning when bed time is and how to fall asleep on his own. I think you're doing well with your comfort after 10 minutes at this stage.
Do come get him when he wakes up during the night and cries at this age. Feed him, make sure he's dry, get him comfy again, walk away, let him cry a little. If your demeanor is confident and kind and not guilty and scared, it will matter a lot to his ease. He will know, "oh, it's time for me to sleep, everything is fine".
As he gets older, be sure he eats a lot ALL DAY LONG. Babies won't over eat, but they will often seem fine when they could eat more. This makes them wake up hungry at night. The "On Becoming Baby Wise" routine of sleep, eat, wake did wonders for us as far as making the kids feel secure sleeping without eating. We would feed them immediately whenever they woke from a nap or in the morning, and let them dose off after being awake, not feeding them to sleep. A feeding before bed is still OK and needed. We did end up adding some extra feedings to get them full through the night too.
Once they are fully fed and used to sleeping without eating or rocking etc, they make it through the night by 3 months or younger. Take it from my step sister-mother of 12. They all slept happily alone by a few months old. Full belly (from the whole day, not just right before bed), good night routine, then walk away. He's way too young to manipulate you by crying. Worry about that later-but with a good foundation, it won't come to that! Good work, good luck!