Do it step by step and don't rush yourself. I almost gave up when I purchased my first DIR-628. I got it at a liquidator for very cheap! Anyways, when my initial configuration experience went sour, I went to the manual and read only the relevant parts, especially upgrading firmwares and such. You must adhere to what DLink recommends, by not closing the browser, or refreshing the browser.
Some suggestions:
Did you unpack the zip file, when downloading the firmware?
Did you make sure that nothing was running in the background?
I know these might sound like silly questions, but they have an impact in these matters.
If you haven't done so, there is the direct official link to the firmware file:
ftp://ftp.dlink.com/Gateway/dir628/Firmware/dir628_fw_124NA.zip
Copy and paste to your browser address bar. Then unzip that file so that you can upload the firmware.
BE WARNED!!, more than likely your configurations might go back to factory defaults once you've updated to any new version of firmware. You then will have to reconfigure your settings again.
I hope this piece of information helps:)