All spell-checking requires looking up words against a collection of other
words. This collection is usually, but by no means exclusively, in
dictionary files of various formats.
Completely client-side spell-checking would involve the dictionary being
available client-side. Since all modern browsers are designed specifically
to prevent any direct interaction with the hardware and software of the
machine on which they are running, this dictionary would need to be
downloaded with the HtmlResponse as some sort of JavaScript array - hugely
impractical as well as massively resource-hungry.
A far better solution would be to use an AJAX-based solution. Obviously,
you could roll your own, but there are loads of 3rd-party solutions
available. I'd suggest you use one of those. I use this one:
http://www.essentialobjects.com/Products/EOWeb/SpellChecker.aspx