Hi Tina
I recommend you call the ABA Helpline 1800 MUM2MUM for some suggestions, but generally if you want to increase your supply, it is a good idea to feed more often (think of it as a temporary time investment - stay in bed for 3 days and feed feed feed!!! - get your DH and family to help you out with everything else for this time.)
Re how early to start - I'm not sure if anything can be done until your milk actually comes in, but once again, someone at the ABA should be able to help you out with this.
I sometimes fall asleep when feeding too - I think it is to do with sheer exhaustion and the hormones that are produced when you're feeding that relax both mum and bub. Have you tried feeding while lying down, making sure that bub is safe?
HTH.
Let me know if you want any more info - I actually have some stuff about this topic from the ABA at home (I really should read it thoroughly...)