The article document class and most other document classes I'm familiar with use the counter enumiv to number the items in the thebibliography environment. To "format" this counter for bibliographic labels, LaTeX by default employs the instruction \@arabic\c@enumiv in two instances in the definition of the thebibliography environment.
Hence, to change the numbering style of the labels from arabic (1, 2, 3, ...) to alph (a, b, c, ...), you could do a redefinition of the thebibliography environment in your document's preamble. Both instances of \@arabic have to be changed to \@alph. In addition, it's necessary to redefine two further macros, \@bibitem (because in its default definition it is hard-coded to use arabic-style numbers) and \biblabel (to change the delimiters around the bib labels from square brackets to round parentheses in the bibliography section).
\makeatletter
\renewcommand\@bibitem[1]{\item\if@filesw \immediate\write\@auxout
{\string\bibcite{#1}{\theenumiv}}\fi\ignorespaces}
%% to replace "\the\value{\@listctr}" with "\theenumiv"
\renewcommand\@biblabel[1]{(#1)}
%% to replace "[#1]" with "(#1)"
\renewenvironment{thebibliography}[1]
{\section*{\refname}% %% use \bibname instead of \refname if using the
%% report or book document classes
\@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%
\list{\@biblabel{\@alph\c@enumiv}}% %% instead of "\@arabic\c@enumiv"
{\settowidth\labelwidth{\@biblabel{#1}}%
\leftmargin\labelwidth
\advance\leftmargin\labelsep
\@openbib@code
\usecounter{enumiv}%
\let\p@enumiv\@empty
\renewcommand\theenumiv{\@alph\c@enumiv}}%
%% instead of "\@arabic\c@enumiv"
\sloppy
\clubpenalty4000
\@clubpenalty \clubpenalty
\widowpenalty4000%
\sfcode`\.\@m}
{\def\@noitemerr
{\@latex@warning{Empty `thebibliography' environment}}%
\endlist}
\makeatother
Finally, to change the delimiters of the citation "numbers" (now: letters!) in the text of the document from square brackets to round parentheses, I recommend you employ the cite package and issue the following commands (still in the preamble, naturally):
\usepackage{cite}
\renewcommand{\citeleft}{(}
\renewcommand{\citeright}{)}
Happy LaTeXing!