I would agree with Robert Hue, that you can't really contact them that easily. Now technically if they are not using their host's proxy service to hide their contact info, they are supposed to fill it out with correct information. Otherwise fake information (including fake proxy info) can be sent to ICANN via their WhoIs complaint form, but that's just 1 possibility. Even if the information is wrong, reporting it to ICANN may not lead to the results you want, but it is an option if the information is blatantly fake or you contact someone other than one of the contacts listed (using the WHOIS info).
Another possibility is assuming the information is correct, but they don't want to respond to you. Again, nothing you can do about that.
Unless you plan on taking legal action against that person, there isn't much you can do to pressure the domain host or owner into giving you any response.
Your best option is if they allow the domain to expire, generally you can purchase the domain after it's expired for 30 days, but that may depend on the domain host and any additional packages they may have purchased to extend that grace period (NOTE: I'm not 100% sure you can extend the grace period, just putting that out there). However, after it's expired it may be more expensive to start, but once you purchase the domain you will have control over it as long as you keep it renewed.
There isn't any other situation I can think of off the top of my head, but as long as they renew the domain, they have full control over it. So you may have to be patient or use a different domain until it opens up.