How do I use the 'Profiles' tab in System Preferences? I've found a bunch of random (sketchy-looking) files inside the Profiles pane of System Preferences. 

To delete them, I need to be an admin, but no matter how many times I put my account on admin, it says I don't have admin permissions. My thinking is that  one of the configuration profiles is causing this.

Is there any way I can find/write a configuration profile to make myself an admin so I can delete the rest?
Good thinking - although they seem sketchy - they are likely well designed and well placed by professional IT. Are you sure you don't know how they arrived or any context about someone enforcing some restrictions or providing some software / benefit / services in relation to this Mac?

The longer answer why you can't really get around this is likelyd ue to those profiles being placed there by an organization / MDM and Apple backs up that enrollment process by getting positive consent before the "locked" profiles are accepted by macOS.

The correct manner is to get the company / person / MDM team to revoke / remove / modify the restrictions that have been placed.


They could remove the restrictions entirely. 
They could push less restrictive parts to profiles.
They could say no.


If they say no, then you could ask - if you're not OK with their controls, would it be possible to wipe the device and then use it with different controls. They might have it enrolled in DEP - device enrollment program - which would then prompt to be re-enrolled like iOS Activation Lock, so be sure you exhaust all other options on a personal level before you resort to an erase.