As a DACA student, am I eligible for PhD funding? I am a deferred action student, meaning that I apply for in-state resident tuition but I am not a US citizen (non-permanent resident). 
I have a work card that allows me to work like any other citizen and pay taxes accordingly.
However, due to me being a non-citizen, I'm concerned that I may not be able to receive funding for PhD. My family does not have a lot of money and we spent a lot of money for my undergrad education (I don't qualify for FAFSA either) so without financial aid, I'm not able to attend graduate school. Even with my part-time job, most of my earnings went into paying for GRE and application fees.

Do university departments offer financial aid to non-citizens (but not international) as well? Most of the departments I saw state that they offer guaranteed funding through the 5th year of a student's PhD but do not explicitly say if it's available to students such as myself.
The most common kinds of funding for PhD students in the US are research assistantships and teaching assistantships.  Typically neither of these types of funding have rules about the citizenship status of the student.  These types of funding would provide a DACA PhD student with tuition and a stipend for living expenses.  

Some prestigious, higher paying fellowships offered by the federal government would not be available to a DACA student.  For example, the NSF GRFP requires permanent residency.  The NDSEG, being funded by the military, requires citizenship or national status.  

On a broader note, if you cannot get financial aid including a stipend, you should not get a PhD.  If the university thinks you will be a good PhD student, they will be willing to pay you.  This advice does not apply to professional degrees.