Is there a standard contract to have a private caregiver sign? I already performed a background check. This all is new to me.

Here are some websites with examples of caregiver contracts that you can download or fill out online.

https://eforms.com/employment/independent-contractor/caregiver/

https://www.wonder.legal/us/modele/caregiver-agreement

https://www.payingforseniorcare.com/caregiver-contract.pdf

https://www.medicaidplanningassistance.org/personal-care-agreements/

The best thing to do is to see an Elder Care Attorney as he/she can make sure that the caregiver contract is completed correctly.

Given that there is this sort of information on every search engine I think you can just google this. As with, say, a rental agreement, there are many forms right on line. Google Caregiver Contract examples. Hope someone else has some they can give you site for.

What are the living arrangements for your uncle? Is he living by himself? If someone else is living with uncle then that would be an indication as to what time the caregiver got to the house every day.
You can have a list of things that need to be done. List them in order of priority keeping in mind that your uncle is at the top of the list.
A log book where the caregiver leaves notes could also be used a sign in/out paper.
Or you can have them text you when they arrive, but doing that you can not be sure of where they actually are when they text you.
Putting a camera in the house would be a sure way to know when they arrive and would give you the ability to check on things once in a while. Cameras are legal, recording is legal but not audio unless both consent to it. But check local laws.

You need to get in touch with an elder law attorney to prepare the contract so all is legal n your state and so Medicaid benefits and requirements are not at risk.