Regardless of how you end up doing this, you are going to need a source for real-time stock prices. Perhaps Yahoo has an RSS feed?
As far as YOUR php site updating frequently, you can try a few approaches
1) create an iframe on your page that reads your data source and has a meta refresh tag so that it updates itself every 5 seconds or something
2) Create an AJAX script that reads your data source
3) Use Flash to read your data source.
A database is probably not necessary for this...you just need to find a data source you can use and that will dictate what approach you need to take.