In order to qualify for Head of Household, you must meet the requirements listed here Do I qualify for Head of Household? 

And to qualify for Child Tax Credit, you have to claim a qualifying child that meets the requirements listed here What is the Child Tax Credit?

When you check the links that AndreaC1 provided, you'll see that one of the requirements for "Head of Household" is that you paid more than half the cost to run your home.

Also, please be aware that when parents live together only one parent may claim the child for all tax benefits. The parents cannot choose to split these benefits.