There is a website devoted to permits and obtaining them.
https://www.nps.gov/yose/planyourvisit/wpres.htmPermits are self registration walk-ins from November thru April. The rest of the year, you compete with others for space in the trailhead quotas. You will also need a bear canister, one of the types on the list at the Yosemite website, in which all your food, trash, and hygiene items must fit every single night you are out. (Bears are active year round. No bears actually hibernate - the ones in California often get up and wander around, and some of them never bother to nap in winter at all, given the abundance of trash that people leave around.) These can be rented when you pick up the permit, which you are required to do in person. Applications for reserved permits are done online (link on page above).
Since you will be unable to park overnight on Tioga Pass after mid-October, trailheads in the valley and on the southern end of the park will be your options, unless you can time your start date with the weekend YARTS bus and be dropped off. You should also be well prepared for below-freezing temps at night, some winter weather (it can snow any time of the year at higher elevations, but it's far more likely late September - April) and be aware of elevation sickness symptoms and understand how to mitigate those/when to depart instead of "toughing it out."