Everything is done through their website. There are a few Assistant City Carrier districts still hiring. They are usually posted with varying numbers in the post, such as Philadelphia Assistant City Carrier 509, 612, 907, etc etc. If you see something like that around where you are that means there is a good chance you can at least take the test.
The assistant positions currently are not career positions, if hired you will basically have a year to prove you are worth keeping on. If not you will be let go.
The site guides you through the entire process. You will get emails telling you about the next steps. There is a lot of paperwork involved, and numerous deadlines that must be met to advance to the next part. You will take the test, results will be emailed to you by the time you get home. If you pass 70+ [but realistically you want to be in the 80's at least unless you are a veteran).
Mandatory Drug tests, as well as background and driving record checks will be next. All on a strict deadline, the drug test was free for me. Not sure about other areas.
A note, you cannot drive for the USPS within two years after having your license reinstated from a suspension. Meaning if your license was suspended, and you got it back on August 20th 2011, the earliest you can drive for the post office is August 20th 2013.
Unfortunately they neglected to mention this to me until after I took the interview and the postmaster I spoke to basically told me I would be hired as I meet all requirements. It wasn't until almost 3 weeks went by and I got no notification that I emailed the woman who was in contact with me and she told me I couldn't drive for them until November of this year. (for a parking ticket in front of my house nonetheless!] This is after I went through their entire system, filled out stacks of tax forms and aced the interview. I was told I would be getting an email from them, which never came. This was all in April.