To modify the number of probes or the probe intervals, you write values to the /proc filesystem like
echo 600 > /proc/sys/net/ipv4/tcp_keepalive_time
echo 60 > /proc/sys/net/ipv4/tcp_keepalive_intvl
echo 20 > /proc/sys/net/ipv4/tcp_keepalive_probes
Note that these values are global for all keepalive enabled sockets on the system, You can also override these settings on a per socket basis when you set the setsockopt, see section 4.2 of the document you linked.
You can’t “check” the status of the socket from userspace with keepalive. Instead, the kernel is simply more aggressive about forcing the remote end to acknowledge packets, and determining if the socket has gone bad. When you attempt to write to the socket, you will get a SIGPIPE if keepalive has determined remote end is down.