i haven't done it, but from the descriptions and pictures i've seen, the mountaineers route is not exactly the ideal place for a first backpack trip - there are certainly a few folks around here who have done it & can give more precise suggestions & ask more precise questions of your preparedness, but for starters:
how much experience do you have with ice axe, crampons, steep snow, steep granite, exposed routes, elevation?
are you in pretty good shape?
i think you should consider doing Mt. Langley first and see how that goes unless you are pretty solid in most if not all of the above skills or have really exceptional good luck. And/or if you have time for a multiday trip, you can probably still get a permit to come in from army pass to the south & maybe exit via whitney trail after climbing to the top.