You poor thing. I agree that the feeding is possibly caused by overtiredness & trying to soothe himself to sleep, and the poor sleeping (especially in the day) is almost certainly caused by overtiredness.
My evenings with DS used to be like this although he was ok at feeding in the day.
I would definitely get hold of a sling & try to give him at least one nap a day in that. Don't try staying in & jiggling around (that was my mistake at first) but go straight out for a walk. It may take him a few tries before he gets used to it, but slings will often persuade the most difficult babies to nap. And buggy naps, while annoying and hard work for you, will give him a bit more daytime rest, so it is worth trying to do as much as you can of those. The more he sleeps in the day, the better he will get at sleeping both day & night.
Don't worry about trying to get him to sleep on his own in the cot - cuddle/rock/whatever to sleep and put him down when deeply asleep (after 15-20 mins is usually a good bet). Or try co-sleeping / lying down with him, even for naps.
A dummy may help him to get to sleep. Sucking is very important to some babies to soothe them into a state of drowsiness, and his feeding pattern sounds like he might be one of them (my DS is). You may well have to hold it in for him - DS (15 wks) still hasn't got the hang of holding one in for long but it really helps him. Try different shapes / sizes as babies like different types, and keep trying it even if he doesn't seem to like it at first. It took DS a good while to get the hang of being soothed by the dummy.
Don't worry about getting him into bad habits at this stage - you need to get over the overtiredness, and if you can sort that, his sleep (and hopefully feeding) should improve. The more he sleeps in the day, the better he will sleep at night. Things will almost certainly improve with the sleep as he gets older and more able to self-soothe.
Try to start getting him off to sleep 45 mins to an hour after him waking up, or else it will get harder as he will be getting overtired. DS will now feed to sleep if he is feeling tired, but won't if he's not yet tired - and he didn't use to if he were overtired.
Try looking for tired signs as well as watching the clock - they can include yawning, slowing down, getting quieter, losing interest in surroundings, fussing, grizzling or crying. However, some babies don't show any signs. DS didn't until he was a bit older.
Don't wake him for a dreamfeed (not sure if you're doing that or not) as you need him to learn to stay asleep.
As GetThePartyStarted says, 6 weeks is also classic growth spurt time. Ours was not much fun - fussy feeding and then yelling because DS wanted to sleep but didn't know how to. Things may calm down a bit once you get past this stage.
And remember: This will pass! It used to take me up to 2 hours to fight DS to sleep each time but now we've got past the overtired stage I can feed him to sleep in 15 mins and he's gradually getting the hang of going to sleep on his own.
A bit of an essay, but hope it helps.