This is probably the easiest way up Swisshelm, at around 6 miles RT and 2900' of gain. Other possibilities exist, including ridges to the north and south, but I'll leave that for someone else to explore!
This was our route up Swisshelm, from the NW. It's probably the most straightforward route up the mountain at present. (click to enlarge)
Green = paved road
Light green = easy dirt road
Yellow = high clearance/4WD dirt road
Orange = road walk
Red = bushwhacking/cross-country travel
See directions on the main page.
From where you parked, look for the faint road dipping into the drainage to the south, and follow it. After popping up the other side, continue on the road as it heads mostly south to round a ridge spur before heading up the lower slopes of the range. At one point, the road splits, with the main track appearing to go left; however, stay on the fainter track to your right. You'll know you took the right path if you see a faint sort of 4-way intersection shortly thereafter (stay straight).
As you continue up the road, you'll see why parking was advised where it was! The road is narrow, rocky, overgrown with catclaw, and offers virtually no places to park or turn around. It does however offer an easy avenue to the base of the steep stuff. At 5400', there will be a faint road heading right and traversing the hillside, quickly ending in a sort of gravel/talus pit in a gully. Take it--now the fun begins!
Cross the gravel pit in the drainage and begin heading uphill on the other side. The brush is moderately thick here, and progress may be a bit slow. Ahead is a bunch of catclaw, which is a pain (literally!) to get through, though most can be dodged if you're careful with route-finding. Once past the catclaw, continue uphill to a sort of shoulder on the ridge above, and if the catclaw has discouraged you, take heart--it actually keeps getting easier from here on out!
Atop the shoulder, the ridge briefly abates, but then continues up at a moderately steep pitch pretty much all the way to the ridge. The vegetation lightens up, and rocky obstacles are few and can be clambered over or easily avoided, depending on your preference. Finally, 1400 vertical feet after leaving the road, reach a cliff band that is your final obstacle blocking the way to the crest of the ridge. Simply sidehill and bypass it on the right, and you'll finally top out on the ridge and start getting some nice views toward Swisshelm, and to the Chiricahuas.
Turn right and pretty much just follow the ridge until you can go no higher! You'll go over two minor bumps on the ridge--maybe 100' tall at most--which are honestly easiest just to go straight over. At times, there are faint game trails following the crest of the ridge that will ease progress, but you can do fine without them. The true summit is hidden until you've done most of the uphill, but when it pops into view, it's only a few easy minutes away. Reach the top, and enjoy the views!
For the descent: we took a slight variation that may make your life easier on the way down, but could be hard to find on the way up. Once you're back on the steep ridge heading down, and have reached the minor abatement above the catclaw, head somewhat more NW (to descender's right) and follow the relatively-catclaw-free slopes down to just before an obvious drainage. Don't try to get into/cross the drainage, but stay just above it on easier ground. If you can find the faint segments of trail/path here, it will make progress through this area fairly easy, and take you back to the gravel pit, from which you can rejoin the road.
Long pants, sturdy shoes, sunscreen, water