I've got a Linksys firewall/router and two Linux hosts, each running Apache. I'd like one of the web servers to host my personal sites, and the second web server to host my clients' sites.

A little more specific...
192.168.1.1 (Linksys gateway w/ external IP)
192.168.1.2 (mysite1.com; mysite2.com; mysite3.com; mysite4.com)
192.168.1.3 (client1.com; client2.com)

The specific problem I'm facing is how to forward port 80 to both hosts and have only the specific host respond? Is this even possible with a Linksys router?

The only solution I've found is to host all the domains off a single host and then redirect my clients' sites to client1.com:8081. This will send the traffic back out on the Internet and from there I can forward port 8081 to my second host.

Any better way of doing this with my existing hardware?

You could use apache's proxy feature to keep it all running on the one port but there is no way to use 1 port and 1 ip addresses for 2 servers.