Since the weight of the bike is carried from the wheel to the forks by that axle, it will be necessary to take the weight of the bike by some other means, e.g. a front stand. Rickety sketchy means of supporting the bike are asking for the bike to come crashing down, potentially doing lots of damage. Do the job right, with the right tools. Buy yourself a front stem stand (not the kind that goes into the bottom of the forks). There are other ways to do it ... in my workshop, I have hooks tied into a steel beam that is a structural part of the building which I can hang tie-down straps from, in order to hang things that are awkward to support otherwise.
With that done ... Remove the nut from the one side, and the axle can be gently tapped and pushed out through the wheel and then pulled out the other side. It's easy. For what you are doing, you need not remove the wheel ... a piece of wood underneath the wheel to stop it dropping out will keep it roughly in place for re-assembly (so that the brake disk does not drop out from between the brake pads, and so that the spacer on each side doesn't drop out).
Having a front stem stand will allow you to take the front wheel off for future tire changes - this will save you money in the long term (the workshops charge less if you bring them the wheel on its own as opposed to bringing the whole bike). Tip, on this bike, the brake caliper need not be removed ... you can sneak the wheel out with the caliper still in place.
