I wouldn't mess with the laptop display but a computer monitor can be a decent display for an xBox1.
This is the display I'm using in my bartop build hooked up to a 19" computer monitor running an early Coinops 5 Lite beta- crappy cellphone pic:
The skin is widescreen squashed to 5:4 which I'll fix later on.
You have two main options that will give you a very high quality image.
hardware only solutions- any xbox
find an LCD monitor or TV that supports Component (green, blue, red RCA) video. I don't think s-video or composite are worth messing with after comparing them to Component.
Get a high quality component to VGA converter
Get one of the VGA output timer boards that converts from xbox to vga + jamma. I bough one of these for testing and will be throwing it on eBay soon.
software + hardware- xbox 1.0-1.5
The cable lilshawn linked to is an option if you don't mind soldering. It will require a bit of software change using Frostythesnowman's (refered to as frosty most of the time) sofmod patch or hard mod bios to properly output. I personally find that native 1280x720 via VGA is the best looking solution if your monitor doesn't support the same resolution via component cables.
I made my own just to try it but I doubt I will do it again since I have found easier solutions.
I will be adding the LM1881 and other items in the link lilshawn posted just because I want to mess around with hardware based scanline generation but the etra hardware is not required at all as long as your monitor supports sync on green and you don't care about SLG. Basically any monitor that has DVI as well as VGA will support sync on green. I picked up the monitor in the pic above from goodwill for $12 after finding it on this list of monitors that support sync on green:
http://ps-2.kev009.com/sog/If you get a monitor that supports sync on green and you don't want to mess with SLGs,
it is super easy to do what you want.
get a standard xbox component cable + this $1.90 cbale from monoprice:
http://www.monoprice.com/products/product.asp?seq=1&format=2&p_id=6025&CAWELAID=1329452164&cagpspn=pla&gclid=CJnBkISqkrMCFUZgMgod62kAqg+ this $.68 coupler from monoprice:
http://www.monoprice.com/products/product.asp?c_id=104&cp_id=10430&cs_id=1043009&p_id=3002&seq=1&format=2With a frosty patch on your softmod + your xbox (1.0-1.5) set to allow 720p + a sync on green monitor, you are done. I have personally done this and it looks great.
Frosty makes a boot disc too. If you boot via this disk, you can try out the VGA mod for just that session. If you have a 1.0-1.5 xBox, I say make an order from monoprice and give it a try.