Just before I start - can I just clarify the situation - as I want to be clear about the wife and kids:
You - Nigerian citizen, illegally in the UK
Your wife - Brazillian citizen, legally in the UK on tier 4 visa.
Child one (13 yo) - ? citizen, legally in the UK as dependant of wife?
Child two (6 yo) - ? citizen, legally in the UK as dependant of wife?
Ok - you're right - you really do need a solicitor for an article 8 application as if it's not carefully prepared it's likely to be turned down initially and have to go to appeal.
AFAIK - you need to make the application on FLR(O) form (which covers all other application types) with a strong cover letter, and evidence of the family, kids and the fact you're their father etc. The children are strongly in your favour.
However, since your wife is not on a visa leading to settlement, you may struggle to prove that you have a established family life, because her course and visa here may come to an end at some point and she would then need to return to Brazil with the children, and this would be a known/accepted risk for her and the kids. This means they might have cause to refuse article 8 as she isn't making the UK her permanent home yet.
If you are living hand to mouth (and care barely find the HO fees) - how is it you are going to prove that she has sufficient funds for the new student visa?
http://www.ukba.homeoffice.gov.uk/study ... /how-much/#
http://www.ukba.homeoffice.gov.uk/study ... ts/family/#
It looks like she needs to show 2 months funding - which is £1200 and for two kids - a further £800 - meaning £2000 in the bank.
M.