I'd probably be a horrible assistant since I tend lately to be offline for days at a time, but if I can help, I will.
I'm not sure, in this context what the difference is between "try to build something, or find something and add to it." If you find a pre-built computer that's already good for classic gaming, great! Building's main advantage IMO is you know exactly what's in your rig. The comp I had before Mazinkaiser gave me issues because I didn't know exactly what motherboard or bios I had and thus could never actually look up the documentation for it, which is something you might conceivably need to do... for example if you're buying RAM for it and need to know what kind of RAM it takes (you can't just stick any old ram stick in there--it has to be within a certain speed range).
Just ask me if there's anything else you're concerned about. If I don't know, then I know people who do. Actually the people over at Vogons might help too-- http://www.vogons.org/ . Though be prepared for a lot of people to say "just use Dosbox."
Although if all/the majority of older PC games you wanna play are DOS games to begin with, then maybe Dosbox is the better option. Like I said I went with building mostly due to classic Windows games that don't work/haven't been patched to work with modern Windows versions. Although according to the review series Ancient DOS Games, there are a few MS-DOS games that don't play nice with Dosbox--Dungeon Keeper being one of them.
EDIT: Also, like the guy above me said, you don't want to go too crazy with getting a classic monitor. Getting a ridiculously old one risks it coming pre-damaged and possibly not supporting the higher display resolutions of your video card (quite irritating to have a game that requires 800x600 then be saddled with a monitor that can't go higher than 640x480!) Also, I personally don't swear by CRTs--my Mazinkaiser has an LCD, though I switched to a CRT for awhile, and noticed SOME differences but not enough to make me prefer one over the other (save for one... the LCD I have sometimes needs to be re-adjusted because the graphics of a game will go off screen or be weirdly positioned. This tends to happen in Native DOS mode a lot and with older titles, and it could easily be due to the monitor and not anything inherent to LCDs).