How the author name could be made italic in citation using biblatex?

You are able to change the formatting of all last names that appear in your citations

\renewcommand{\mkbibnamelast}[1]{\mkbibemph{#1}}

...and then reset the formatting before \printbibliography so that the names appear in upright font in the bibliography. Your MWE would become:

\documentclass{article}
\usepackage{filecontents}
\begin{filecontents}{jobname.bib}
@book{ALEXANDER1977apl,
    Address = {USA},
    Author = {Alexander, Christopher and Ishikawa, Sara and Silverstein, Murray},
    Date-Added = {2010-02-24 11:14:54 +1100},
    Date-Modified = {2010-06-01 12:10:48 +1000},
    Publisher = {{Oxford University Press}},
    Title = {{A Pattern Language: Towns, Buildings, Construction}},
    Year = {1977}}
\end{filecontents}
\usepackage[backend=bibtex,indexing=cite,style=authoryear,citestyle=authoryear,sorting=none,maxnames=1]{biblatex}
\makeatletter
\AtEveryCite{%
  \let\bibopenparen=\bibopenbracket%
  \let\bibcloseparen=\bibclosebracket}
\makeatother
\addbibresource{jobname.bib}

% Renew formatting of the last name to add emphasis
\renewcommand{\mkbibnamelast}[1]{\mkbibemph{#1}}
% Revert formatting of the last name for bibliography
\AtBeginBibliography{\renewcommand{\mkbibnamelast}[1]{#1}}

\begin{document}


\noindent Text citation: \textcite[pp 45]{ALEXANDER1977apl}

\printbibliography

\end{document}

End result: