This answer extends Philipp Lehman's method for generating a possessive labelname using a format alias. It should work for all variants of the standard author-year styles.
Note that the solution is bad for plural labelnames, where the possessive is formed with just ' and not 's. This case is probably very rare, though. Other languages may not be as easy to handle. For an example refer to a previous question, which considers Norwegian possessives.
\documentclass{article}
\usepackage[style=authoryear-comp]{biblatex}
\usepackage{filecontents}
\DeclareNameFormat{labelname:poss}{% Based on labelname from biblatex.def
\ifcase\value{uniquename}%
\usebibmacro{name:last}{#1}{#3}{#5}{#7}%
\or
\ifuseprefix
{\usebibmacro{name:first-last}{#1}{#4}{#5}{#8}}
{\usebibmacro{name:first-last}{#1}{#4}{#6}{#8}}%
\or
\usebibmacro{name:first-last}{#1}{#3}{#5}{#7}%
\fi
\usebibmacro{name:andothers}%
\ifnumequal{\value{listcount}}{\value{liststop}}{'s}{}}
\DeclareFieldFormat{shorthand:poss}{%
\ifnameundef{labelname}{#1's}{#1}}
\DeclareFieldFormat{citetitle:poss}{\mkbibemph{#1}'s}
\DeclareFieldFormat{label:poss}{#1's}
\newrobustcmd*{\posscitealias}{%
\AtNextCite{%
\DeclareNameAlias{labelname}{labelname:poss}%
\DeclareFieldAlias{shorthand}{shorthand:poss}%
\DeclareFieldAlias{citetitle}{citetitle:poss}%
\DeclareFieldAlias{label}{label:poss}}}
\newrobustcmd*{\posscite}{%
\posscitealias%
\textcite}
\newrobustcmd*{\Posscite}{\bibsentence\posscite}
\newrobustcmd*{\posscites}{%
\posscitealias%
\textcites}
\begin{filecontents}{\jobname.bib}
@Article{bertram:sa,
author = {Bertram, Aaron and Wentworth, Richard},
shortauthor = {BW},
title = {Gromov invariants for holomorphic maps on Riemann surfaces},
journaltitle = {J.~Amer. Math. Soc.},
volume = {9},
number = {2},
date = {1996},
pages = {529--571}}
@Book{title,
title = {A Book with No Author},
date = {1933}}
@Book{shorthand,
title = {A Book with No Author},
shorthand = {ABWNA},
date = {1933}}
@Book{label,
label = {Fallback label},
date = {1933}}
@Book{does,
title = {A Book by the Does},
author = {Doe, John and Doe, Jane},
shortauthor = {{The Does}},
date = {1990}}
\end{filecontents}
\addbibresource{\jobname.bib}
\bibliography{biblatex-examples}
\renewcommand{\baselinestretch}{1.25}
\begin{document}
\subsection*{cite with labelname}
posscite many authors: \posscite[3036]{aksin} \\
textcite shortauthor: \textcite[e.g.][3]{bertram:sa} \\
posscite shortauthor: \posscite[e.g.][4]{bertram:sa} \\
textcite shorthand: \textcite{kant:ku} \\
posscite shorthand: \posscite{kant:ku} \\
textcite compact: \textcite{aristotle:physics,aristotle:poetics} \\
posscite compact: \posscite{aristotle:physics,aristotle:poetics} \\
textcites: \textcites[3036]{aksin}[529]{bertram:sa} \\
posscites: \posscites[3036]{aksin}[529]{bertram:sa} \\
posscite with plural labelname (where this solution is bad): \posscite{does}
\subsection*{cite without labelname}
textcite shorthand: \textcite{shorthand} \\
posscite shorthand: \posscite{shorthand} \\
textcite title: \textcite{title} \\
posscite title: \posscite{title} \\
textcite label: \textcite{label} \\
posscite label: \posscite{label} \\
\end{document}

\citeauthor{texbook}'s \citeyear{texbook}should work? – Seamus Jul 4 '11 at 13:17