If you want to climb Apache and Navajo in a single day, this is your route! It works well in either direction, but I'll describe it as a climb to and not a descent from Apache'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 Navajo Peak. This would involve a climb of Navajo's Airplane Gully route and a descent of its fourth-class West Chimney 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.
The most difficult move is the first one. From the base of Dickers Peck, climb a short broken face to attain the south ridge's crest. From here, climb either on the ridge crest, or stay slightly to the south on some third-class ledges. The scrambling is quite fun! As you trend higher, you'll reach Apache's 13,400' subpeak. This subpeak has two summits. You can skirt the second on its west side, but don't try to skirt the first because of some steep drop-offs. Descend to the wide saddle south of Apache, and climb easy class 2 talus to the summit.
If you're climbing Navajo Snowfield to reach the Apache/Navajo saddle, crampons and an ice axe are a must!