You can't install splunk on your router, unless you are using a Linux server as a router, in that case it is possible.
I assume you want your router logs to be send to a Splunk server. In that case the router need to support syslog to send out the logs to a syslog server.
The best way is to let your router send the logs to a server, let a syslog deamon (syslog-ng, rsyslog) catch the messages and write them to a file, and then let Splunk monitor that file. That way if you are fooling around in Splunk and need to restart Splunk you don't lose you logs for that time period. You can run the syslog deamon and Splunk on the same server (for home use...).