Hi OP, first - it sounds like you’re doing a great job and much of what you describe is normal and not necessarily a sign that your baby isn’t getting enough milk from you and needs top ups.
I would really recommend trying to follow an EASY routine which stands for Eat, Activity, Sleep and something for You.
So, first the baby Eats. Breastfeed for as long as the baby wants and offer both sides each time.
Once feeding is over, the baby should not be fed to sleep if possible but be awake. This is the Activity time in the routine. This is the time to burp the baby if necessary and then this is when nappy changes should happen. If the baby had fallen asleep whilst feeding, this will wake her up. At this age, this might be all the awake time that your baby can manage but as they get older, this is when you will do tummy time and interact with each other, start to play etc.
Once the Activity time is over (and it might be just 30 or so minutes at this stage, including the feed) your baby needs to go back to Sleep. The baby is likely to become pretty cranky and this is when you might wonder if they are hungry. However, if they’ve had a full feed, take comfort from this - the problem shouldn’t be hunger. Babies can become very upset when they need to sleep! A 6 week old will not be able to stay awake for very long - their “wake window” will probably not exceed an hour and may be less - 4-5 hours is waaaay too long - so if your baby is fed, has a clean nappy and has been awake for 45 minutes to an hour, the overwhelming likelihood is that crying is due to needing to go to sleep. At this age, it is likely your baby will want to sleep on you. If you don’t have one, I’d really recommend a sling. If your baby cannot yet go to sleep in their own cot/Moses basket etc, pop them in the sling and they should go to sleep within a few minutes. Your baby might prefer being pushed in the pram - work out what they like. I use a dummy at this stage as babies love to suck something to soothe themselves to sleep. Remember, at this age they do not know how to put themselves to sleep and need to be taught. It is quite rare that you can simply put them down in their cot and they will go to sleep on their own. You could try patting them, white noise etc. Try a few different things and work out what they like. I recommend not putting them in a darkened room during the day as you want to encourage them to distinguish between night and day as much as possible.
In theory, once the baby is asleep you should have a bit of time for You. This is certainly possible if the baby is in a pram/cot/Moses basket but you might be a bit more limited if they are in a sling. They will probably now sleep for a couple of hours, maybe 3/4.
So, you then keep following this same pattern on repeat (though it’s not quite the same at night: you do not do the Activity part at night, save for changing the nappy where needed). A 6 week old will probably need to feed roughly every 3 hours, but it may sometimes be 4 and may sometimes be 2. Try not to think of this as a schedule but a routine - where the length of time between each cycle may vary but the pattern is always the same.
Occasionally, there might be a need to feed again briefly after the Activity time. On the whole, this should not be needed and my philosophy is to try and follow this routine but if the baby won’t settle after a few minutes of trying to get her to sleep, it’s possible that she may need a little more milk to get her there. Once fed fully, sleep will then come - but I would give it 15/20 minutes of trying to get the baby to sleep before you decide that more milk is required and, if possible, offer more breastfeeding as opposed to formula as you can’t overfeed a breastfed baby.
As for expressing, I would try not to put too much pressure on yourself here. If you follow the routine above for a few days, the baby should respond and fall into it. If the baby’s nappies fill okay and she gains weight, I would not wake her for feeds, but feed her when she wakes up. If she sleeps longer than 3/4 hours, that’s great (so long as she’s filling nappies and gaining weight)! If you do want to express, I would recommend doing this a little time after you’ve fed the baby so that your milk has recovered post feeding and so that you have milk for the next feed if needed. So, maybe after the Activity time and whilst the baby sleeps. You will get more milk in the mornings and at night but I really do not recommend pumping at night - get all the sleep you can then. And I certainly would try to avoid pumping more than once or twice a day - any more just puts so much pressure on you. You will have more milk if you are well rested and not stressed.
I also really recommend Lucy Webber’s Instagram for tips. She also has a book about breastfeeding and the ‘fourth trimester’. Babies pulling off the breast, writhing around, grunting, shouting, bashing with their hands is all normal at this stage. This will ease as she gets a bit older. This stage is just really tough but none of this means you are doing something ‘wrong’.
For tons of useful advice on everything else, I’d recommend reading The Baby Whisperer, which is where the EASY routine comes from. If you try the routine, make sure you persist. Within just a few days your baby should respond if you stick to it. I hope this is helpful. Good luck.