Most of the home routers, whether they have one, four or eight ports, can handle up to 255 devices connected to them. The four or eight ports on the router are actually on a switch that's in the same case as the router.
Buying another router to connect to your router adds a additional, unneeded complexity to your LAN. You *could* do it that way. It would isolate the computers coming off the second router from your LAN unless you configure the second router to act like a switch. (Of course that would raise the question as to why bother with a router if you're going to configure it to just act as a switch anyway?)
Or you could replace your 4 port router (actually a router with a built-in 4 port switch) with an 8 port router (actually a router with a built-in 8 port switch). But the only things you would accomplish by this would be you'd only have one power supply, and one case sitting on your desk. But you also loose a little flexibility. For example, if the computers are located in two clusters, with a separate router and switch you could run a single cable from one cluster to the other cluster. With a single 8 port router, you might have multiple cables running parallel to each other from that single router.
In general, I'd say that it's usually preferred to use both the router and the switch than it is to buy a new router.