You need to look at chapter 4 of publication 463.
You can't use the standard mileage method if you don't own the car, because the standard mileage method includes depreciation and you can't claim depreciation if you don't own the car.
If you use the exact expense method, you can't claim depreciation for the same reason. You can claim other expenses according to the rules for the actual expense method. Basically, you need to record all the car expenses for the year -- all the gas, all the insurance, all the maintenance, all the repairs. You also record all the mileage, business and personal. You figure the percentage of business miles, and you can deduct that percentage of the actual expenses, provided you paid for the expenses. So you could include insurance, if you paid it, but only for the percentage of the car mileage that was business use. If you didn't pay for all the gas, all the repairs, and so on, then you can't include those items in the expense total when figuring the deductible percentage based on business use.