To start I'd go to a site like Priceline.com and enter Tempe, Arizona for the town and 5 miles for t

To start I'd go to a site like Priceline.com and enter Tempe, chicago downtown hotels Arizona for the town and 5 miles for the search radius (Tempe is just to the east of the airport, most of the Tempe car rental places will be closer than many Phoenix places).
I don't have an address but we rented from Hertz offsite from a car dealer. We were dropping the car off in Las Vegas and Hertz did not charge chicago downtown hotels extra for a one way rental. We took a cab and I remember that the dealer was close to the airport. If you check the Hertz website, you can check the locations and find the office that is closest to the airport.
There are many answers to this very question and I have previously reported on my favorable experience with Enterprise (off site). Just type "Phoenix car rental" in the Search the Forums box and you will pull up many threads on this very subject.
4. Take the (free) shuttle to the hotel/motel, spend the night there and have Enterprise pick you up there the next morning. They will do this for you free of charge. Of course, your reservation must be with a branch location that is closest to your hotel/motel. This is the Enterprise location chicago downtown hotels we used:
6. At the conclusion of your rental period, repeat the same process without the additional one night's hotel/motel stay. Better yet, you may be able to persuade your Enterprise driver to drop you off at the airport itself. We did! (While off-site chicago downtown hotels car rental branches can't pick up at the airport, they can and do drop off if asked discretely.)