First thing to do is to add a field for the journal abbreviation to the article entrytype, journalabbr in the MWE. Therefore you need to declare a datamodel (an extra file; in the MWE I've used filecontents to simulate that) and you have to tell biblatex/biber to use it in the package options.
Then you have to modify the cite command, so it fits your needs. \citep from the authoryear-style (your citestyle) uses the \cite command. There you just have to add a switch, which checks if the field journalabbr is empty or not, and prints it out or not.
Last but not least you have to add the abbreviation fields to the entries.
MWE:
\begin{filecontents}{min.bib}
@article{boisson2003unexpected,
title={Unexpected protein families including cell defense components feature in the N-myristoylome of a higher eukaryote},
author={Boisson, B. and Giglione, Carmela and Meinnel, Thierry},
journal={Journal of Biological Chemistry},
journalabbr={JBC},
year={2003},
publisher={ASBMB}
}
\end{filecontents}
\begin{filecontents}{authorjabbryear.dbx}
\ProvidesFile{authorjabbryear.dbx}
\DeclareDatamodelFields[type=field,datatype=literal]{journalabbr}
\DeclareDatamodelEntryfields[article]{journalabbr}
\end{filecontents}
\documentclass[fontsize=11pt, paper=a4, ngerman, DIV=calc]{scrartcl}
\usepackage[scaled]{helvet}
\renewcommand*\familydefault{\sfdefault}
\usepackage{fixltx2e}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{babel}
\usepackage[german=quotes]{csquotes}
\usepackage[style=authoryear-comp,sortcites=true,sorting=nyt,isbn=false,natbib=true, citestyle=authoryear,bibstyle=authoryear,backend=biber,maxnames=1,maxcitenames=1,
,datamodel=authorjabbryear%added!
] {biblatex}
\DefineBibliographyStrings{ngerman}{ andothers = {{et\,al\adddot}} }
\renewbibmacro*{cite}{%from authoryear.cbx
\iffieldundef{shorthand}
{\ifthenelse{\ifnameundef{labelname}\OR\iffieldundef{labelyear}}
{\usebibmacro{cite:label}%
\setunit{\addspace}}
{\printnames{labelname}%
\setunit{\nameyeardelim}}%
\iffieldundef{journalabbr}{}{%
\printfield{journalabbr}%
\setunit{\nameyeardelim}%
}%
\usebibmacro{cite:labelyear+extrayear}}
{\usebibmacro{cite:shorthand}}}
\addbibresource{min.bib}
\begin{document}
\citep{boisson2003unexpected}
How it should look like:
(Boisson et al., JBC, 2003)
\printbibliography
\end{document}
Best Answer
The problem was that biblatex isn't included in MikTeX 2.8. It was solved by updating to 2.9.