WARNING: If you are not BOTH interested in the fine points of halacha (Jewish law) and a coffee drinker, you will no doubt find this post incomprehensible, irrelevant, or both. But for the observant coffee drinkers out there — you might find this useful as a way to enjoy a better cup of coffee on Shabbat than instant!
If you want to make coffee using a French press on Shabbat, there are two issues to be concerned with: borer (sorting), and bishul (cooking). We'll treat them separately, the easier one first.
Borer, sorting, allows you to take the desired (coffee) out from the undesired (grounds). With the French press, you are pouring out the good coffee, and it leaves the grounds behind. Filter coffee is potentially problematic on Shabbat because it appears you are selecting out the bad instead.
Bishul is potentially an issue to be concerned with. First, of course, you can't boil water on Shabbat, for that is cooking; so you have to use water that comes out of your urn that was turned on before Shabbat.
The issue of concern is whether you are "cooking" when you put the coffee and water together. My first thought was "ein bishul acher bishul" there is no cooking after cooking (you can't violate "cooking" for something already cooked) and since coffee is roasted it's cooked. But there is a principle that "yesh bishul achar afiyah" there is cooking after baking, and coffee is roasted, hence baked not "cooked" which means with liquid. Not everyone agrees that there is bishul after afiyah. The Shulhan Arukh quotes both opinions without expressing an opinion (318:5); however, even if you go with the more stringent opinion that there is, this is still not an insurmountable problem. As long as you pour the water into the press first, and then add the coffee, it is a kli sheni, and generally speaking we hold you don't have to worry about cooking in a kli sheni.
There are those who might be machmir and argue that since the coffee is "afiyah" (baked) it should count as "kalei bishul" easy to cook, and hence susceptible to cooking even in a kli sheni. If you take that position, you would need to first pour the water into a different cup and then into the press so that it's a kli shlishi. I don't hold with that opinion, however, since there are reasons to doubt whether there is bishul achar afiyah, and we have reasons to doubt whether roasting coffee makes it kalei bishul, there is no need to adopt both stringencies.
No need to suffer with instant coffee on Shabbat!