The actual climb begins at the lake just west of the Whitney-Russell Col. The most common approaches are: (1) hiking up the N. Fork of Lone Pine Creek, or (2) in conjunction with bagging Whitney, directly off the summit plateau of Whitney. In either case the start is from the Whitney Portal trailhead.
When approaching up the N. Fork of Lone Pine Creek, you start up the main Whitney Trail. After 1/2 a mile or so, a turnoff onto the NF use trail is made, and the use trail is followed to Iceberg Lake (see the route info on the NF trail). Cross-country travel then is used up the obvious route to the Whitney-Russell Col. This approach even with a light load can still take up most of a day, and so many will find it helpful to camp at Iceberg Lake.
When approaching from the summit of Mt. Whitney, the main Whitney Trail is taken to the Whitney summit. From there, proceed down the north face of Whitney via a series of chutes and ledges. There are ways to keep this route 3rd class if good route-finding is effected; however, ease of descent will also depend on snow and ice accumulation. Expect a few 4th-class-type moves and occasional rappel-gear sightings along the way. When snow and ice is at a minimum, the same chute to return to the Mountaineer's Route can be used to drop off the summit plateau with little difficulty --- just aim for the lake between Whitney and Russell.
Once the base of the south face of Russell is gained, take the large, obvious chute of small-to-medium-sized talus up towards the summit crest of the peak. You will be to the east of the Fishook Arete, and to the west of the Whitney-Russell Col. This chute narrows, and eventually meets a headwall just below the crest. Turn left, and scramble up a 3rd- and 4th-class gully for a hundred feet or so until the headwall (on your right) is reduced to a couple moves in the 5.0 range. Up to this point, you have basically been following the path of least resistance, with no obscure routes or tricks, and this is no different. Make a couple of super-easy edging moves with huge holds, and you're done! Well, kind of, as long as you can handle lots of exposure on the narrow summit ridge. Once on the crest, turn LEFT (west) to get to the highest point. Make your way around blocks for about 30-40 yards and BINGO! Reverse the same route OR, for the fast way back to the Portal, take the East Ridge down to the Russell-Carillon saddle and cruise the sandy talus downhill to Clyde Meadow.
Bob Burd adds: "While [this route] can be done with plenty of class 4-5 variations, careful route-finding can keep it to no more than exposed class 3. " What he means here is that a variation of the south face (to the left of this one) does not require the 5.0 exit moves that this route follows.
Winter climbing conditions may greatly differ from this description, so pack accordingly! 4th/5.0 section is relatively short, so a full rope length is not necessary. It is unlikely that on Russell itself any long rappels would be needed in any season.