Liverock is the filter for marine tanks. What you need is a skimmer, which will lift particles out of the water before they rot, a heater (hooked up to a temperature controller - google STC-1000), and lots of flow - for that you need powerheads, which need to be sized around 30-50 times the display tank volume.
What'd be better though, is to drill that tank, and sump it. There are lots of DIY projects to look at where people have done this. It's not as scary as it sounds, and will give you more volume, greater stability, easy access for maintenance, and a place to store the ugly kit.