A lot of different "alternative biologies" have been considered, but if you analyze their required chemistry, this always falls apart at some point, some mechanisms necessary for life are simply impossible. Replace carbon with silicon, increase ambient temperature, and you're getting a lot of interesting reactions that could be conductive to life. But instead of CO2, which is a gas, you have SIO2 which is a crystal. You can't breathe crystal. It's awfully insoluble, so no simple hemoglobin counterpart. And so on. Won't work.
Water and carbon based biology is one most plausible. There are odd variants, like bacteria thriving on sulfite, manganese and copper oxide, but they still require water to live, and still use carbon as the fundamental building block.
While others might exist, chance for them to form is much lower than for one similar to ours. And since we haven't designed even one fully plausible alternate biology that doesn't involve water we don't even know what sort of detectable signatures it would produce. So we search for something we know might yield results; hint at possibility of life somewhat similar to ours - instead of a wild goose chase after the unknown.
Actually, there exist one "alternate biology" which is fully plausible even if its early origins would be Earth-like: Silicon based, artificial intelligence, robotic physiology, electrical power, proliferation through fully automated manufacturing processes. It's unlikely to produce any detectable chemical signature (at least distinguishable from half of the natural celestial bodies out there), but we are fairly sure its electromagnetic spectrum - radio waves signature - would be quite specific. Well, The SETI project is on it.