Is it possible for me as a graduate student in the US to a sign a contract with an US company to create a US patent from my PhD research?
In general, the procedure works roughly like this:


Optional, depending on how potentially valuable this situation might be: Contact an intellectual property owner of your own, especially if considerable monetary amounts are involved.
Optional: It can be very helpful if you can see if anyone in your department has had experience with this personally, as you can go to them and they can often provide a few insider tips and suggestions that would be more relevant to your local situation.
Contact your University's Intellectual Property Office (or equivalent) to schedule a meeting to discuss the situation. Initially you just attend yourself with any co-creators, company isn't involved yet at this stage.
If you are in a place where the University claims ownership, you'll need to file something like a discovery statement outlining the intellectual property. If the University doesn't have ownership, they'll often be willing to discuss the option to assign it to them as part of a royalty agreement. If they do have ownership, as the creator it is still typical to receive royalties as part of the process (it is not ownership that is strictly free of responsibility on their part), but refer back to the IP office and your lawyer for specific guidance.
Your assigned representatives (lawyers, generally) will try to work out a contract with the company to define the transfer of rights, limitations, royalties/payments, etc. 


As you are new to the process, the only way anything is likely to work out well for you is to get experienced people on your side to look after your rights (and ensure they themselves get paid for doing so). 

If you are working with a University of any significant size and experience - and just having a PhD program suggests the place probably qualifies - there will be an office of people who have done with this many times before and can advise you better as to your specific situation. 

There is huge variation in the IP agreements you signed between institutions and local laws, so I believe what I outline above is as close to a universal answer as possible. The devil is always in the details, so make sure you talk with people who are familiar with the local laws and policies that will apply to you.