What's the best way to get the "Rampage!" achievement?

I'd suggest playing easy bots and being more sneaky than usual. I got this achievement on my 2nd try versus easy bots on the Baggage level. First off, I stuck to the outside of the map (right side from CT spawn ramp) and peaked bots to kill them and retreated when necessary. Definitely do not go running around the middle of the map, try not to get shot even once because your health needs to last the entire match.

It is very possible to go a long while without getting shot once versus easy bots. My first try I had gotten shot once for 13 damage all the way until nearly the end when some bot got lucky and killed me.

The last kill with the knife was pretty tricky... The bots all had shotguns at that point and it was difficult to get close enough to kill one without getting shot. I would recommend going down the ramp from CT maybe into the lower baggage area and try to find one that isn't looking. Be Very sneaky when trying to get the final knife kill, the knife kill may take a few tries.

If you wanna do this versus humans you're just going to have to be VERY good and get lucky by playing against some weaker opponents. I really think that against humans this is nearly impossible. Stick to easy bots. (If you want to cheese it, bots_knives_only will probably get you an easy win, but you said you wanted it legitimately)

I'd recommend having only 1 enemy and 2 allied bots; (bot_kick )

With this setup your allied bots will find and distract the enemy, possibly lowering his health, while you go in for the actual kill.