I remarried last year but I provide ALL of the expenses of my household. My spouse does work but it is my income that pays all the bills, and the upkeep of the home. Can I still file as Head of Household?

If you are married, your filing status can only be Married Filing Jointly or Married Filing Separately unless you did not live with your wife for last 6 months and meet certain conditions.
If you have income and your spouse does not have income or has a very little income, then it is better to file as Married Filing Jointly. If both of you have almost equal income, then it won't make much difference. Also may credits and deductions are not available if you file Married Filing Separately. So normally you should file as Married Filing Jointly, unless there is a compelling reason to file otherwise.

As Married Filing Jointly, you get one extra exemption deduction of $3,400 and one extra standard deduction of $5,350. Thus your total deductions (if you don't itemize) are $17,500

Filing HoH is NOT a option for you In my opinion.

MukatA's points are accurate and well-spoken.