Hi C.-
Congrats on your baby boy! What you are describing sounds totally normal to me. If he sleeps 5 hours in a row at 5 weeks old, he's doing amazing. Both of my kids did not do that! My daughter slept maybe 3 hours in a row and my son was up an hour after he was fed. And yes, when they are awake, most newborns do not have a long period of being happy or quiet. I found my kids as newborns to wake up and be happy for maybe 15 minutes. Then they became fussy and cried a lot and finally went to sleep. I thought something was wrong with my kids so I put them on the hypoallergenic formula Alimentum. It didn't make much difference except that I felt better because now they couldn't be allergic to it.
some things you can do to help your baby - you can try to structure his day by having him feed, play and then sleep. Play at this age means looking at a picture for a few minutes or having you sing to him, looking out the window. After he eats he's going to try to fall asleep. Jostle him awake for the play time. Sometimes it will work, sometimes not. Just try. Do not let him stay awake for more than 1 1/2-2 hours at a time. So, if he's up at 7:30am, his first "nap" will be by 9:30am. So, let's say he eats at 8am. At 9:15am, swaddle him, rock him until he's really drowsy and then lay him down to fall asleep in his crib if that's where you will want him to be sleeping. If he cries, pick him up and try it again - rock and then lay down. Do not let him sleep past his next feeding time which will be around 11am. You want him to learn daytime. Expose him to the sunshine. Wake him by unswaddling him and laying him down. He'll stretch. Feed him around 11:30am and then his next nap will be around 1pm. Do the same as before. Swaddle, rock and lay down in crib. If it doesn't work, try again. Be sure he is up for his 2:30 feeding. Repeat the cycle again but don't be surprised if now he becomes more alert and when you put him in for his nap around 4:30pm that it is a shorter one. This nap and the catnap at 7:00pm can be taken in a swing, car, or other location and not his crib. Be sure he is up for his 5:30pm feeding. Let him have a catnap of a half hour around 7:00 or so. Then do an evening routine with him - bathe him, carry him around, look at things. Feed him around 8:30-9:00pm. Play for a few more minutes and then put him to bed. He will wake to be fed. This is normal. He will cry and fuss and generally not sleep well until he is a few months old. I do not feel that ADHD has anything to do with newborn behavior. I would not consider that at this point.
You need to catch up on your sleep and you'll feel better. When he naps during the day, I know you have a million things to do. Lay down and rest when he first goes to sleep for his naps for at least an hour. Then do what you need to do. When he goes to bed at night, you must as well. I remember being tired enough to actually go to bed at 9pm! Let me know if I can help you more.
Good luck!