If you want to climb Navajo and Apache in a single day, this is your route! It works well in either direction, but I'll describe it as a climb to -- not a descent from -- Navajo's summit.
The base of the climb is from the Navajo Peak/Apache Peak saddle at 13,100'. There are two main ways to reach this saddle, but both require some effort. The first way to reach this saddle would be from over the summit of Apache Peak. From the summit of Apache Peak, descend it's South Ridge route. The other approach could be called the more direct, but it requires a climb of the steep Navajo Snowfield that rests just northeast of this saddle. This is a serious climb on snow that reaches a maximum angle of 45 degrees.
From the notch between Dickers Peak and Navajo, climb southwest for 100' up an obvious third-class ramp. From the top of this ramp, you can see the west chimney, a 100' cut in Navajo's west face. This is the northernmost gully on the west face. When dry, the chimney is third-class except for one fourth-class move in the middle of the climb. According to Gerry Roach, the chimney contains steep snow in June. Have fun climbing the 6-foot wide chimney! Holds are abundant and you'll be finished with the climb long before you want it to be. Once you exit the chimney, you'll find yourself on accomodating terrain below Navajo's distinctive summit cliff. Look to the right and pick out the platform on the skyline south ridge. Traverse to the right (west) under the summit cliff and then south, reaching the platform. You've now joined the Airplane Gully route. Scramble north, and the summit is yours!
According to Gerry Roach, the chimney contains steep snow in June. If you're climbing in late-spring/early-summer, crampons and an ice axe are a must!