Few minutes before 12:01 am PDT Friday, open 2-3 browsers and open Apple.com. Better yet, install Apple Store app on iPod touch or older iPhones (if you can).
Around 11:58 pm PDT Thursday, start refreshing the page and iOS app every 15-30 seconds. Apple has occasionally delayed iPad and iPhone pre-orders by 15 minutes or so in the past, so don't be alarmed.
When pre-order shows up, fill out the information across multiple browsers and iOS app one after another. Upgrade eligibility page is where pre-order has been problematic in the past, so try one browser after another as needed. In my experience, iOS app is the most reliable and least troublesome as it transfers the least amount of data AND is least used for pre-order.
Actual shipment varies depending on where you live, but if my past experience anything to go by, you should receive your iPhone 5s around Wednesday.