Xbox – Copy Xbox 360 saves to Xbox One without Cloud

You will lose all your save game data.

Profile information including achievements, arcade purchases and DLC content will come with you. You'll need to download them again, obviously. Any media content you've purchased through Zune, or the like, would be recoverable as well.

A possible solution for getting your save data transferred is to use one of many techniques to bring your existing Xbox back to life. Do a search and you'll find many solutions -- some involving towels, or pennies, or major surgery on your Xbox. Do this once you have a new Xbox and have USB stick formatted for Xbox use.

Once your old Xbox is (kind of) working again you can quickly transfer items to the USB stick. Not everything can be moved though... some games are jerks. :(

Have you taken these steps (from MS Support)?

Install a game on your hard drive

1.Insert the game disc into the disc drive.

Note After you insert the disc, the default action is to automatically start the game. If this happens, exit the game and return to the Xbox Dashboard.

2.From the Home hub, select the game in the drive tray, and then press X on your controller.

3.Select Install.

4.A progress indicator appears while your game is being installed on your hard drive.

Note Installing an Xbox 360 game might take 12 minutes.

The disk is still required to play (needs to be inserted into the Xbox360).

Here is another question listing benefits/downsides of installing games.