steve,,, The millermatic 135 requires a 20 amp 110v circuit correct?.... so if you plug anything else into it while welding at full power you would be exceeding the rating of the circuit...
Also as you weld over time the wire can get hotter... so yes it would be easy to overload your typical 15 amp circuit...
Look at the miller and see what the draw is and then confirm what circuit your house wiring is listed as....
if you begin blowing the breaker and resetting it and continuing to weld over and over you do risk fire,,,, ususally the circuit breaker will trip repeatedly and stop you from doing this...
know your machines draw and the power supply ability of your house wiring....
if you have to you can pull a dedicated circuit just for your welder...
In my shop each machine has its own circuit.... and the outside ones are on GFI breakers to protect me when wet.... of course welding and water dont mix
I recommend "Uglys book of electrical wiring: ,,, you can find it at most electrical supply houses and maybe even the depot... its what the pros use... about 9.00
hope this helps
cam