Not sure what you're asking. You want to use your vita to remote play with your friend's ps4? You cant. First, you need to link his ps4 to your vita, and then you need to have it auto sign into your account. Unless you are account sharing aka game sharing with your friend, you cant remote play with your friend's ps4.
wait what? So to remote play I have to be signed in with my account on his PS4? I have to be at his house to register?
When you first try to link the ps4 to the vita. You will need to be at his house. Go to remote play on the vita, click on remote play and your vita will say, searching for nearby ps4. When it detect his ps4, it will show up and you link it. After that, you dont need to be at his house anymore since its already linked, you can remote play from anywhere (from your house to his house) as long as his ps4 and your vita have internet connection. But in order to use remote play, I believe his ps4 must be sign into your psn account.
edit, forgot to mention, you will need his ps4 numbers when linking. Go to his ps4 setting, connection to vita setting, add device, and will show his ps4 numbers. Enter them on your vita to link. Only need to enter once, after that, you can remote play anytime.