In your library you may have a print edition of an old and good essay by Isaac Asimov, #"The one and only"#, see (http://www.madsci.org/posts/archives/2001-06/993247450.Ch.r.html) which addresses your question. (FYI Asimov was a very celebrated and prolific science fiction writer who formally trained as a biochemist).
Carbon can form up to 4 covalent bonds, and it can form reasonably strong bonds with nitrogen, oxygen, and other heteroatoms. Carbon also has an almost unique ability to catenate: to form long chains with #C-C# linkages, and along the chain there can be regions of different reactivity (by the presence of functional groups).
In addition, a #C-C# bond is unaffected by water, and thus, if say a cell, can be isolated from the aqueous medium (and membrane formation is considered to be a pre-requisite in abiogenesis), an intracellular chemistry can be developed distinct to that of the the outside environment. If such cells can self-replicate, by drafting in and chemically modifying atoms from the outside environment, you have got more or less a #"living system"#.
Carbon chemistry thus allows the possibility of large and complicated molecules and polymers, which can demonstrably support a biochemistry.
