Mt Tunnabora (4135m) is located on the eastern shore of Lake Tulainyo,
just east of Mt Russell.
First you need to get to Lake Tulainyo (the highest named lake in the USA). One option is to follow the sandy southern route to Russell's east peak, i.e. the Russell-Carillon pass. The other option is much better although way less popular. Take the Whitney mountaineering route just like going to Russell; then leave the trail before the first lake and hike straight north, parallel to the creek that you passed on the way up. This route is shady, foresty, it has water and it is class 2; much better than the steep sandy slope to the Russell-Carillon pass. Eventually you will converge towards the creek and you will realize that you have been hiking towards a massive mountain, which is actually one of the lower subpeaks of the Cleaver. The creek drainage winds its way upstream into a gulley: first the drainage bends left, then right.
Entering the gulley between Carillon and Cleaver:
To avoid brutal bouldering the right side becomes more appealing as the soft grass disappears. Avoid the steep granite sections. The mountain to your left is Mt Carillon. The gulley between Carillon and Cleaver gets narrower but never narrow enough to call it a "chute". It is obvious that it dead ends against the real southern face of the Cleaver. Before the dead end, there is a climbable "cliff" and a chute next to it. You can either climb the cliff (with a little exposure) or enter the chute (class 2, lots of scree, but safer). Once into the chute take the first sensible chimney to your left that will take you to the top of the "cliff".
The route up the gulley (cliff or chute):
The chimney to the left of the chute that takes you the top of the cliff:
Here you realize it is not a cliff but simply the bottom of a broad class-2 face. Ascend this face and you'll get to the ridge that separates this part of the world from the Lake Tulainyo part of the world. The crossing is obvious (not the lowest point but just south of it).
The location of the Cleaver-Carillon pass if you are coming from Lake Tulainyo:
The Cleaver-Carillon pass if you are coming from Lake Tulainyo (note the vertical stripes of Carillon):
The descent towards Lake Tulainyo is relatively easy despite the bouldering. Most people prefer to walk around the lake using the southern and western shores, but you need to do the opposite: get down to the lake, then move right (east, counterclockwise) and after about 500 meters you should be walking on sandy beach or snow (depending on the season). Ascend the sandy slope to your right: that's Mt Tunnabora.
It is a fairly uneventful ascent. If that shore of the lake looks impassable because the water level is high, then you need to walk around the lake clockwise and ascend Mt Tunnabora from its western ridge, which is even easier.
Note: as of 2008 the unelected officials who run the Inyo National Forest (with your tax money) have extended the "Whitney Zone" to the entire Whitney mountaineering route above the Lower Boyscout lake. Hiking beyond the first lake requires a Mt Whitney permit (no, this is not a joke). The route that i described here should lie entirely outside the Whitney Zone, but the Inyo National Forest keeps expanding the Whitney Zone (and some day might just declare independence) so i can't guarantee this will still be legal when you read this. Petition your Congress representative to dissolve the Inyo National Forest and return the wilderness to wilderness lovers.