Just wanted to know what would happen if my biological father filed taxes with himself as the head of household and me as a dependent and I file the taxes as the head of household with no dependents. I'm 22, unmarried, not living in the same state or have the same permanent address, earning more than him and have been providing for myself.
If you provided for yourself and made enough money, you do not qualify as a dependent. That's an important first point.
If your father files taxes and claims you as a dependent, that's an invalid filing and tax fraud.
If you have no dependents, you can't file as head of household (that would also be fraud).
If you claim yourself as single and not a dependent, and your father claims you as a dependent, both of your returns should be flagged as invalid - though, depending on the ordering and timing of the filing, one may go through and later require amendment (or be considered fraud and possibly warrant an audit).
To file as head of household, you must have at least one "qualifying person" as part of your household. In general, that person must have lived with you for "more than half the year". (With some exceptions, like if the person was away for military service or college.) As it appears that you do not live with your father, then you are not a qualifying person to enable him to claim head of household status.
If you have no dependents, you probably don't meet the rules to be a head of household either. The qualifying person doesn't necessarily have to be a dependent, there are a few other cases, but they must have lived with you for more than half the year, and you have to be paying over half the cost to maintain the home.
See IRS 1040 instructions, page 16. https://www.irs.gov/pub/irs-pdf/i1040gi.pdf