To the best of my knowledge, there two broadly plausible possibilities: one operating at very low temperature, the other at very high temperature.
The somewhat less plausible route is to use silane (and silene/silyne) compounds, in as-direct-as-possible analog to carbon compounds. Silanes are not as stable as carbon chain molecules, and so would need a colder environment. This means you need a cryogenic solvent, as well- water won't do. Silica, the direct analog for carbon dioxide, dissolves well in various organic solvents (which are used to manufacture, e.g., silica aerogels) with low freezing temperatures, but if there are organic solvents around, you'd be much more likely to end up with cryogenic carbon-nitrogen life instead, or maybe a hybrid that occasionally incorporates silane molecules into a primarily carbon-based chemistry. So, yeah, you need to pretty much eliminate carbon altogether, and that means, most likely, using an ammonia-water mixture as the primary biosolvent. (A mixture of water and ammonia has a lower melting point than either liquid alone.) Like organic solvent, ammonia will react with oxygen, which means you can't have much free oxygen in the atmosphere, so these organisms would need to be hydrogen breathers, or use some other alternative metabolic cycle- perhaps one that doesn't require breathing at all, and relies entirely on simple decomposition of energy-rich molecules.
The more plausible option is something based on silicone (-Si-O-) backbones, possibly with organic side-chains. Silicones can make conveniently soft and flexible molecules at human-comfortable temperatures (as evidenced by silicone baking ware), but they withstand high temperatures quite well (again, as evidences by silicone baking ware...) and become more versatile at higher temperatures. In this case, there is a reasonable geophysical scenario that would predispose the development of this sort of life over purely carbon-based: a world with large quantities of sulfuric acid, which decomposes hydrocarbons. So, you have a world somewhere between Io and Venus, which gets much hotter than our world, resulting in a loss of water and hydrogen, and concentration of sulfuric acid. Not as small and cold as Io, and not quite as hot as Venus, because you ideally do want to retain some surface liquid, not just clouds. Conveniently, sulfuric acid also reacts with chloride and fluoride salts, producing hydrochloric and hydrofluoric acids, and hydrfluoric acid will dissolve silica, which would serve to make it bioavailable and provide a means of disposing of metabolic wastes.
So, you have a mixture of highly concentrated sulfuric acid with a little water and small quantities of hydrochloric and hydrofluoric acid as the biosolvent, operating somewhere between 50 and 300 degrees Celsius. You don't need to eliminate carbon from the environment--the solvent choice makes the evolution of purely carbon-based life impossible anyway--and you don't want to, because carbon will be useful to incorporate into side-chains and functional groups, just like we incorporate nitrogen into lots of our own "carbon-based" biomolecules. The solvent won't react with oxygen, so there can be plenty of free oxygen (and possibly some chlorine and fluorine) in the air as well, although there may be little motivation for purely oxygenic biosynthesis (whether photosynthesis, chemosynthesis, or what have you); you could instead end up with a mixture of free oxygen and a bunch of sulfur trioxide in the air, either of which could be used as oxidizing agents for animal metabolism. (In our environment, sulfur trioxide is a solid, but it's a gas above 45 degrees Celsius.)