I think the classic 5D is a bit too dated... you can get such good deals on a 5D II that I wouldn't bother with the classic 5D.
On a budget your best choices are likely the 6D or the 5D II.
You should be able to find a used 5D II in good condition for easily under $1000 (likely somewhere in the $700-800 range).
A used 5D III is quite a bit more....
KEH.com is showing the used value to be closer to $2000. (In contrast... a "new" 6D is just under $1300 for the "body only" and a "new" 6D II (which only just started shipping) is $2000 for the body only.
The 5D III offered moderate gains over the 5D II -- I think the most notable difference is the new focusing system (the 5D II used the old 9 point AF arrangement). It offered slightly higher resolution. Slighter better ISO performance, etc. Basically the 5D II is still a pretty nice camera at a bargain price. The 5D II wasn't a good choice for action photography (it doesn't have a fast burst speed, it doesn't have an advanced focus system, etc.).
The 6D is also a good choice for entry-level full-frame and frankly that's what it's designed to be. You get a fantastic full-frame sensor but they cut back on a lot of other frills to keep the camera affordable. It has an 11 point AF arrangement (looks just like the 9 point arrangement but they sneak in two extra points just left and right of the center AF point). Technically it has fractionally better low light performance over the 5D III. The 6D also has a built-in GPS -- which annoys some people because you must remember to disable it via menu system when powering off the camera. It keeps updating it's GPS position (even with camera powered off -- unless you remove the battery completely or go into the menu system and disable it).
Canon did that with the GPS because it takes a GPS a few minutes to acquire a position fix after powering on and they felt that consumers might shut off the camera (to swap lenses for example) and then power it back up ... and then be upset that any photos taken for those first moments would be missing GPS data.
On new models they fixed this by offering two modes... in one mode the GPS is active at all times (even with camera off - just like the 6D). The other mode powers the GPS down if you really switch off the camera ... but if the camera merely goes to "sleep" (power switch still in the "on" position) the GPS will remain on but will reduce the update frequency. The 6D has no optional modes for the GPS other than enable vs. disable. If you don't disable it then it is operating at all times even if the power switch is "off".