The weather in the Eastern Sierra Nevada is no where near as predicable as other areas of California. However, I hiked overnight last year mid-October and found the weather conditions as ideal.
I brought microspikes, but did not need them. There was snow on the ground in spots above ~11k, but could mostly be avoided. Ice was only present where water had seeped from rocks and froze along the switchbacks.
You will need camping gear with cold weather clothes (not Norway cold, but able to handle freezing temperatures). I used a one-person bivvy with foam mat and sleeping bag, along with several layers of clothing. Thermal layering was the best, as I could adjust to the conditions. Wool socks and a cap are nice.
You will likely need a windbreaker for the summit. Mine doubled as a rain/snow coat.
The weather conditions change weekly, so you may want to bring extra gear in your vehicle in case you need to adjust your plans. Keep an eye on reports of conditions here and other sites, like the USFS Mt. Whitney web page.
