Gen Size?

I run a similar welder off of a 4kw Coleman (read underpowered) generator. Works fine up to about 125 amps output, then it's all over. I have not pushed the duty cycle so I'm not sure if the generator would handle any serious welding. (Used for race car repair, a couple of quick welds, anything serious goes back to the shop)
IIRC, the Lincoln is supposed to have a 50 amp breaker to run full power. This would say you need a 12kw unit. This is probably a bit rich since the welder does not have 100% duty cycle at full power. My guess is that you should be able to get full power out with the Lincoln advertised duty factor with about 8 kw continous rated genset.
Keep in mind that the power factor on your welder is not a good match to the windings on the gernerator. A cheap generator may not have the proper circuit breaker and thermal cutouts needed to properly protect the generator if you push it.
Just> I have a Lincoln 225/125 (k1297) stick machine. Does anyone use one of

This is one point where an engine-driven welder shines over a generator and a welder. One which can generate 8KW 110/220AC power can also weld at 250 amps 100% duty cycle (the welder section has it's own set of coils - the 110/220AC section is not involved in welding). Moving two parts would be easier than humping the 600 lb beast around, however.