Halo 2 - 480x640 @ 60Hz interlaced = 9,216,000 pixels/s
Halo 4 - 1280x720 @ 30hz progressive = 27,648,000 pixels/s
Halo MCC - 1080x1920 @ 60hz progressive = 124,416,000 pixels/s

That's a more than 13 fold increase in total pixel throughput in the last 10 years if all you've got now is a 4:3 SDTV.

You might appreciate a marginal increase in image quality since everything will have to be downsampled so much but there will be a lot of that power you're wasting and you'll have to contend with an enormous slew of problems: unreadable text and UI, potentially more lag, extra power, cabling and equipment requirements etc.

All that for an investment of $400 console + $60 game bundle + $?? converter cost, just to play games you already own: total waste of money.

So, if you're going to get an Xbox One, you are going to get a new tv as well.

Now that we've got that out of the way, let's start the 'What TV?' conversation at something like this Samsung. 22" 1080p 60hz LED backlit, recent model, good brand, for under $200. Even has component in so your 360 will definitely connect to it no matter what model you have. Oh, and as a bonus, all your current 360 games will look better than ever!

For new, the market doesn't really go lower than this. You can get bigger or cheaper if you're willing to go for older, second hand models but you may struggle to find them on Amazon.