Across the paved road from the YSI, the "North Rim Trail" starts up the side of the canyon. (It’s paved here.) Climb up it, continuing along the north rim of the canyon. Shortly, the pavement disappears and there is a gate to go around; the trail snakes up and down (but mostly up) towards Eagle Rock (you can recognize it by all the antennae on it). One turnoff to the right is labeled "Loop Trail" -- worth avoiding unless you’re into steep climbs. A couple of spots along the main trail get pretty steep -- you can avoid the second one by staying left as the road forks just at the bottom of that steep part, then taking the switchback right turn that goes up to the top of that part.
Once you get up to Eagle Rock (elev. 790’), take a break. Take some pictures. To get down, take mostly right turns. At one spot, you’re headed West (out towards San Jose) and the trail has a steep branch down to the left, but also continues a lot less steeply, sorta straight. You can see where the steep left branch goes down to another trail. At that point, take that steep left branch -- the other just goes out to a small place where a couple of goats live, and peters out. That steep descent is the trickiest technical challenge on this route.
At the bottom of that steep left branch, turn rightish onto the trail that stays just above the paved road. There’s one nasty down-and-up right turn that’s easy to miss (but if you do, it just puts you out onto the paved road). Once the trail really puts you back onto the road, go right, down the road about 50 or 100 yards, then turn sharply left onto a small trail that takes you back to the main entrance trail, just above those "rough wooden steps" that were right by the start of the ride.
©1995-2019 Doug Landauer
Henry Zaininger (tronmij1@aol.com) adds (1995.09.17):
As an Alum Rock regular I can assure you there are more trails there
than what you put in your description. First, you can ride up the
North Rim from almost the parking lot. Second, when you’re going up to
Eagle rock, if you take the road right at the fork you will come to a
new trail (about a year and a half old). It is called Todd Muir trail
or something. It is a skinny footpath that isn’t too steep but the
downhill part is fun. It will let you out at the mineral springs.
©1995-2019 Ross Finlayson