OK, first of all, if the child is yours only, then your BF can't claim the child as a dependent if you file a tax return, even if you wanted to let him claim your child. If you file a return because your income is more than $12,000, you claim your child and he can't.
Then, "head of household" requires the taxpayer meet 3 tests,
1. be unmarried,
2. pay more than half the cost of maintaining their home,
3. provide care in that home for a qualifying dependent.
Since your BF can't claim your child as a dependent, he can't file as HOH. He will file as single. You could file as HOH if you have paid more than half the total cost of the homes where you lived over the course of the year. (Rent, mortgage, food, utilities, insurance, etc.; including this house but also where you lived before that.)
If you did not pay more than half the cost of maintaining your home(s) over the year, then you can't file HOH either and will also file as single.
