Install the tools you need via the Dockerfile (iostat, top, atop, etc.), and an sshd. Then, instead of running the single web process, run supervisord via CMD, which will subsequently launch both your web app process, AND sshd. From there, EXPOSE 80 22, and you can SSH into the container to run any perf analysis tools as usual.
EXPOSE 80 22; CMD ["/usr/bin/supervisord"]