If you don't need to move it around, build a desktop.
The specs look fine but, for your purposes, building a computer for the same price as buying that laptop would provide you with far more power and would be much, much better suited to video editing and gaming, not to mention that a desktop will probably last you a whole lot longer due to the ability to swap out parts and better cool it. If you don't absolutely need a laptop, then go desktop.
Also, you cannot build a laptop the same way you can build a desktop. At least I've never heard of such a thing, and I'd be surprised if it's been done by anyone other than a few computer enthusiasts here and there.
I would recommend searching for a few more laptop options since that 1366x768 screen is not great. Your laptop screen is the main thing where you will be spending majority of your time engaged in various activities, so you should search for laptops with highest resolution possible that fit your budget. The rest of the specs on that Lenovo machine look good.
Lenovo makes great laptops and that one you've picked will certainly be better than your old PC, but it won't be anywhere near as good as a new PC you could build for $850.
I imagine video editing would be tough on a 15 inch 1366x768 screen.
Does it need to be a laptop? Depending on how/where you're doing your computing, you could probably save some clams by building a comparable desktop PC.
I'm guessing when you say build you mean just choosing the parts on the manufacturers website versus just buying something that is already set and ready to go. Looking at that laptop though it seems to be pretty solid for the price. I wouldn't expect any long term gaming out of that thing though once the next-gen rolls around. You'll get some good mileage on most current gen games.
I don't think you can build a laptop. Manufacturers don't sell laptop hardware by itself, and you'd pretty much have to be working for a manufacturer to be able to fabricate your own.
That seems like a decent laptop but my one concern is the screen, I've heard nothing but poor reviews for 15" screens of that resolution. Plus you really don't want a screen with glare.
I'm in the market for a new laptop. My old 2005 Windows XP (Media Edition) PC is dying out (boot sectors errors). I'm always hearing of building a desktop PC but can the same be done for a laptop?
My other option is to look at buying a Lenovo IdeaPad Y580 Laptop - 20994MU for $849.00. This PC will be for video editing, gaming (Steam), and live video streaming (Justin.tv, Twitch.tv). The specs look pretty good:
Would this be a good buy or would you recommend building my own laptop or looking a different model/brand?
Thanks in advance for your help!