My boyfriend and I have a child, we live together, I am claiming the child this year. Can we both file as head of house hold? Does he also but the child down as a dependent under his name?

Only one of you can be head of household. Only one of you can claim the child.
Read: Your U.S. Tax Return: Filing Status: Head of Household (http://taxipay.blogspot.com/2008/05/head-of-household.html)

You must be unmarried or considered unmarried on the last day of the year to even think about filing HOH.
Further, you must satisfy the five tests outlined here: Publication 501 (2008), Exemptions, Standard Deduction, and Filing Information (http://www.irs.gov/publications/p501/ar02.html#en_US_publink100041781)

It would be impossible for both to file HOH.

One may also note that it is not only impossible for you both to claim HoH, to do so may be construed as tax fraud by the IRS.

People who commit tax fraud pay LARGE fines and sometimes find themselves guests of the federal prison system for a substantial portion of their lives.