How to delete profiles on a used xbox 360 without losing my place in the game I have been...

Which is good. You shouldn't have any of his login since that violates the code of conduct. But in order to continue using your save progress you would have to continue using his profile since you started the game on his account. Otherwise, you will have
to start over on your own account.

Games saves are tied to the profile they are on. So you would have to continue using his profile (which is not recommended) in order to continue your save. So if you were to delete the profile you would lose your save.

So what does it mean when it says" Delete Profile Only – This deletes the Xbox profile but leaves saved games and achievements".....???

I don't have any of his login information, it just shows up on there when I load the game, I haven't been online with the system yet. He had never played GTA5 before and I just started playing after getting the game loaded. He did play other games online
and does now on his Xbox1.

You would still need to sign into the account in order to use the gave save. So once you delete it you would have to be able to redownload it to access the game save.

Ultimately it's best to just start over on your own account.