What chemicals help define "taste"?
It's not exactly true that there are 5 "tastes". There are 5 generally accepted classes of gustatory transducers, each comprising many different types. The perception of taste is a product of not only the output of these transducers, but also information from the olfactory system, visual cues, textures, sound, etc.

The gustatory system itself is based (like the olfactory system) on overlapping sensors with partial selectivity. So a given compound, sucrose for instance, may stimulate several sweetness sensors (and probably even some from other classes) to varying degrees, and the particular combination of the responses is what gives sucrose its unique taste.

The types of molecules that stimulate the different sensors is not known clearly, but we can make some broad generalizations:

Sour - These sensors are thought to be primarily ion channels sensitive to pH—acidic solutions produce a sour taste.

Salty - Also thought to be mostly ion channels sensitive to small metal cations. Different ions have somewhat difference response. ($\ce{Na+}$ doesn't have the same response as $\ce{K+}$) There is some debate about the actual mechanism though.

The rest are g-protein coupled receptors. They change shape when they bind a molecule which causes a chemical signalling cascade.

Sweet - Not known precisely, but many compounds containing hydroxyl or carbonyl groups trigger these receptors.

Umami - Mostly triggered by amino acids, particularly glutamate. Inosinate and guanylate are known to enhance the effect of glutamate.

Bitter - This one is rather strange. A very wide variety of compounds stimulate these receptors, even with very different structural features. This taste is thought to have evolved to protect us against toxic compounds as many poisonous alkaloids and such taste bitter. Denatonium is an example of a bitter compound (one of the most bitter known).

As for whether any other classes of transducers exist, it's entirely possible. Some researchers have identified candidates for receptors for things like fats. Also, there some receptors like TRPV1 (which responds to capsaicin) are known that respond to compounds found in food, but aren't usually considered taste receptors.