Oh dear - that does sound like a mess although if he has been paying tax through the PAYE system recently then that's something, but incurring fines etc with HMRC is a bad thing and he really needs to take it in hand to stop them spiralling out of control. Do you mean that he has outstanding self-assessment income tax returns to complete from previous self employment?
Either way you are in a quandary - either you sort it all out for him and he ends up doing nothing to help himself, leaving you frustrated and possibly resentful but able to claim tax credits, or you leave it to him and possibly miss out on money that you could be entitled to and which could help your family finances.
Assuming he was self employed prior to his current PAYE position, would he have been paid cash in hand? If not, then you could ask his bank for statements to get some idea of when money was going in to calculate his earnings? Or if it was cash in hand, would he had had emails or similar arranging the work? It sounds like a big job, which is probably why he is burying his head in the sand, but could you use a calculator to give you an estimate of how much you might be entitled to in tax credits in order to motivate him? If he thought of that as 'payment' for going through his finances and working everything out, it might encourage him to get to it.
And it might also be worth reminding him that nothing is worse than not knowing - having fines and penalties hanging over his head that could affect your credit for years to come is a nightmare. At least if he sorts it out, works out what is owed and comes up with a plan he will be able to work through it and get it sorted, even if it does take a while.
I hope you manage to persuade him, but if you want more advice then I am happy help as much as possible (mostly from my own self assessment experience and working for an accountant) and there are plenty of knowledgeable MNers who might be able to offer some more specific advice or point you in the direction of more resources that could be useful. Good luck!