God ! The pathetic beginning!!
Heres presenting to u a 'once upon a time' ultra frustrated mum of a now 7.5 month old - or i would rather (if you will) a mirror that shows you the beautiful future 
At 6 weeks....I would gladly kill anybody who said 'dont worry....after 4 months all will be fine'. My reaction was - what are you, kidding me?!?! With as much sleep deprivation and cumulative exhaustion, i would be dead long before its 4 months!
But no, i am alive and much much much happier!
Well...,all the drama above was meant to 1) bring a smile to your face (i hope it did) and 2) a very very heartfelt sympathy. I so remember those first days, weeks, months!
As stated by other mums, the only way to catch up is to forget everything else. Everything means everything and includes every little household work, personal grooming everything. I hope you have some help around for the routine jobs like breakfast, dinner etc. You have to relax all day. I could never really 'sleep when the baby sleeps' but yes, whenever he was asleep, i just relaxed. Watched tv, watched friends on my laptop, took a bath or anything but enjoyed some 'me time'. I also indulged in food 
Also, it was hard but i accepted the fact that for now my 24 hours are the same with no difference between day and night. So for now, just dont bother whether its night or day - round the clock baby will 'need you' every 1 to 2 hours. It could be just for comfort but at this stage comfort is a basic need like eating so please dont ignore this need. Its easier said than done but keeping such little things in mind will really save your soul.
I am one who was (and still is) convinced that i m not cut out to b a mum. But still, today i have a very happy 7.5 month old. And no matter how far it may seem like right now, but this happy point is your future. It might not be just round the corner but let me tell you one great thing - FROM WHERE YOU ARE, IT ONLY GETS BETTER!!