Can you file charges against the police department for entering your home without a warrant?

Yes, you can as long as there was not any probable cause. If they just walk in and start searching, that is illegal and you can press charges. If anything incriminating is found it is also inadmissible in court.

If you committed a crime and ran to the house, or a crime was committed at the house, then yes they can enter with out a warrant, and there is nothing that can be done about it.