Start at the watertanks near the North Face Trailhead.
Note: Bad news!! There has been alot of development around the trailhead. You can still park your car at the gate and you can easily make it to the trail, but there are new no trespassing signs and the HUGE model home that was constructed at the base of the water tanks (info from PellucidWombat). I would suggest asking the WMC for any updates. If you know any other information/tips/updates, please post them here as additions and corrections!!
This was copied from the North FaceTrail route page:
There is no official trailhead to the route, but the trail is easy to find. From I-215, exit onto Wasatch Blvd (northbound; via 39th S, southbound). Drive south and turn left onto Oakview Dr. Follow the road all the way to the end, where it makes a sharp turn to the right and doubles back up the hill a short distance farther. At the end of the road is a gate. Continue on the pavement by foot past the gate. This access road doubles back and ends at some water tanks near the mouth of a gully. Look near the fence surrounding the water tanks and you will see the trail on the east side of the gully.
This is the finest route on Mt. Olympus for snow climbers. When viewed from the north end of Salt Lake Valley, you will notice two prominent snow gullies on the north face. The eastern one is Apollo, the western is one is Zues. Zues is the beginning of the route for the West Slabs. Apollo is the better snow climb.
Despite the low elevation (7,000 feet), there is usually snow in the chute well into July, and even later in some years. May and June are the best months for ascent.
From the watertanks, follow the very steep trail along the ridge just east of the chute. Follow the ridge until is is convenient to drop into the chute. From the valley floor, and from the lower end of the chute, it looks like the chute dead ends into the vertical wall, but it doesn't. It actually makes a 90 degree elbow to the west. Using an ice axe (and maybe crampons), climb the chute all the way to the west ridge of the north summit.
Top section of Apollo Chute.
The west ridge of the north summit can be followed with much scrambling all the way to the north summit. There are several choices for descent. The only practical time of year to make the ascent is in May and June. In summer and fall, there is too much loose scree. In winter there is too much avalanche danger. This is an excellent alternate route to Mt. Olympus.
Apollo Chute is just right of center on the photo. It is only pleasant to climb when snow-filled.
An ice axe is needed; crampons are optional, but can be useful.