I'm posting this answer to consolidate some points raised in the original answers by various people, and expand/clarify points that could have been better-written answers.
Some Useful References
A very interesting book, available for free on the web now, is Xenology: An Introduction to the Scientific Study of Extraterrestrial Life, Intelligence, and Civilization by Robert A. Freitas Jr. This was written in 1975–1979, so it's a bit dated. But, §8.2.3, Alternatives to Carbon, is pretty much what this question is about.
An oft-used reference for Worldbuilding is the Atomic Rockets web site. The page Building Blocks covers alternatives to life as we know it.
The web page Alternative Forms of Life on David Darling's Encyclopedia of Science is similar. It's an index of pages in the same body.
Hot or Cold ?
First of all, let's clarify what “silicon based” bio-molecules might entail. It's well-known that a difference between Si and C is that Si has higher binding strength, and things like silicon grease are produced for high temperature use. So, we conclude that Si analogues of Carbon-bearing molecules would need (and withstand) higher temperatures.
Well, that's not quite right. Our grease and silly putty and caulk contain Si, but are not direct analogues to hydrocarbon molecules. They contain alternating silicon and oxygen as a replacement for carbon in the main chain, and furthermore the side chains are conventional carbon-containing chains.
(( Figure 8.4 in Freitas was made on a typewriter. I wonder if some modern renderings could be done and posted here? Imagine illustration of Polydimethylsiloxane, Phenylsilicone, etc.))
So, high temperature molecules would not be simply silicon based, and still needs carbon. Freitas points out that the increased temperature tolerance is modest and probably not worth it.
First, many silicones tend to disassemble into ring molecules at temperatures of roughly 300–350 °C. (Similar behavior is observed in most complex carbon compounds, but at somewhat lower temperatures.) It would be difficult for silicones to remain stable in much hotter climes, and it is unclear whether this slight thermal advantage is enough to enable Si to out-compete C in a high temperature regime.
On the flip side, pure silicon chains, analogous to carbon chains, are possible at cryogenic temperatures. Furthermore, liquid nitrogen is a suitable solvent. So rather than the Horta, we have Nadreck the Palainian. But, this raises the issue that the energy levels for metabolism would be too high and you would blow everything apart instead of doing the combining and splitting of other molecules that being alive is all about.
So, What Other Elements?
It turns out that complex molecules need carbon as well as silicon. Some useful high-temperature polymers use Si, 0, and H in abundance, and also incorporate some more exotic elements: Germanium, Tin, and Lead for example is shown in Freitas figure 8.4.
If trying to avoid carbon, you can often use half-and-half Boron and Nitrogen instead. Again, Freitas shows this in figure 8.6.
I expected more alternative atoms, like Sulfur instead of Oxygen, to go with the high-temperate idea. But that seems not to be the case, and the molecules we are familiar with use common organic side chains when producing products to work in the 250–300°C range.
However, as Freitas relates, reporting in the mid 1970s about discoveries made shortly before, H. R. Allcock remarked “… it now seems likely that almost any set of required properties can be designed into the polymer by a judicious choice of side groups.”
So we may have all kinds of elements, many more than we have in familiar biochemistry. Instead of an odd metal atom or other oddball as part of a critical catalyst, we might have an abundant number of some other element atoms used as part of the polymer repeating unit, and a smorgasbord of elements in the side chains of useful molecules, as this is how to fine-tune their properties.
It seems to me that this would make it harder to evolve a simple general-purpose toolkit of common parts. Common parts might be based on larger units than familiar life, with several kind of complex molecules all using a large (from our point of view) number of different elements, rather than just the C-H-O-N.
High-temperature life that uses Silicon will use all the same elements we do, and all the other elements it has available as well. Rather than getting varied and carefully tuned properties from different arrangements of a few kinds of atom, this alternate life would need to use many more kinds of elements to achieve the careful tailoring of properties.
The most basic reusable building blocks would be rather large, compared to ours, since it takes a complex molecule with side chains to make it do exactly what is needed. And evolving a new novel distinct molecule for every need, to make a more basic reusable set they need to be general purpose tools that can be reused as-is.
We have a lack of universal solvent. The idea of cytoplasm will be very different, with different ways to deliver, distribute, react these building blocks. That is not elaborated here but ought to be a new question.