Take a deep breath, count to three, then let it out. Hold your baby in your arms and look him in the eye. And repeat after me: I've never been a mother before, you've never been a baby before, so let's cut each other some slack.
Now seriously, relax. Breastfeeding is hard work. Caring for a baby is hard work. Recovering from childbirth takes time, not the least because you don't get to sleep and relax like you normally would; you may feel ok, and so did I, but you're still not 100%. People who make it out as if this was a piece of cake are either lying or very peculiar. It will get better, I promise, but not for a little while, I won't lie to you, so brace yourself and lower your expectations.
My first three months were awful. My baby sometimes cried, sometimes didn't, but mostly didn't sleep. AT ALL. She just didn't, at least not for more than half an hour, or maybe an hour here and there. And breastfeeding was a problem because I had low supply (if you saw the size of my breasts you would fully appreciate the irony of this, they were large before, now they're massive, and still...).
So I had to start giving her formula top-ups. And I had to accept that she would only sleep on me, at the boob and as long as I didn't move. So I accepted it, I really did, and ignored everyone who said that I should keep trying with the cot; this is my baby, I cannot change her, let's work with what we have. Therefore, I would get myself some water, or a cup of tea (or both), and maybe a snack, a box of tissues, the remote, a blanket for her, you name it, and most importantly, I would go to the loo, all in preparation for sitting down for a feed and a nap (for her), while I watched TV. Oh, and I would also get my mobile and the land line close at hand (and on vibration or mute) just in case someone called, because I was not going to get up and interrupt her nap even if the house was on fire.
Then, little by little, things improved. She still would not sleep much during the day, but she started sleeping for two, even three hours at a time at night; we co-slept so she could feed and not wake up if I tried to move her to her cot. She started burping herself (actually, farting) and as a result stopped crying right away. And we also started going out a bit more, not a lot, but some, and I would feed her in a cafe with a slice of cake and a proper cup of tea for me.
Forget what people say, find your own system and keep at it. If your baby only sleeps for short spells in his cot then maybe try co-sleeping and if it works better then do it, safely, of course, but do it. Mixed feeding is not the end of the world either, whatever it takes to fill their bellies is quite fine in my opinion and you're still BF him so that's great (if you stopped it would be ok as well, there are no medals in this game, you just do what works and drop what doesn't, be glad that you at least have options).
If your hair is unkempt, and mine was for months, then so be it; if you look and feel ragged, that's ok too, this is not a fashion show, it's real life and you've just pushed a human being out of your body and are currently feeding said human being round the clock (with your very own boobs, such an odd feeling, isn't it?).
What I mean to say by all this is that I'm sure that you are doing a fab job and you will get better with practice. Just keep at it and don't feel bad about feeling bad, it's ok, really.