If your Xbox 360 is not playing though the Xbox One, this is what you should do, to find and, hopefully remove, the problem.
Test the Xbox 360
Plug the HDMI into your television and see if it works. Try the same cable, first, and the cable used on your Xbox One if it still doesn't work.
If the Xbox works without swapping the cables, everything on the 360 side is O.K.
If you have to change cables, your issue is with the HDMI cable. Try the new cable in your Xbox One.
Test the HDMI in on your Xbox One
Try connecting a different device to your Xbox One. A blu-ray player, your computer, even a set top box is fine (as long as you have HDMI out).
If other devices don't work, you may have a faulty HDMI in port.
If these tests are inconclusive
If the fault has not been found, you may simply need a better quality cable.
Quality does not make as much of a difference with HDMI as it did with component. You are always going to get next to perfect quality - if it works.
Some devices need a bit more power in the cable. Try a cable rated for 4K. Look for gold tips if you want to be sure. "Braided cable" does not improve quality. It just makes the cable less likely to break under strain.
I hope you find your solution here. I can personally vouch for cheap cable, here. The cable I am using cost me ten bucks, and it cost the store $2.
If they are using your account (Which I would never recommend due to security concerns) they could purchase a game doing that on their Xbox One and install it on their console. However the friend will not be able to play the game unless they own it or are using your account. But yes, it is possible to buy games on another console. Honestly it would just be easier to get a card and have you purchase the game instead.
Edit: Another alternative to do this (thanks to Timmy Jim for finding this) is to link both your and your friend's consoles together which would allow you to share games together and also allows you both to play the game at the same time. Here is a link Timmy provided on how to do it.
Source: https://support.xbox.com/en-US/games/game-setup/my-home-xbox
Best Answer
From the OP, originally posted in the question body: