You will need to redeclare the article driver as the code to insert the title is directly in that part:
\RequirePackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{Diederich2006,
author = {E. Schweizer and A. Hoffmann-R{\"{o}}der and J. A. Olsen and
P. Seiler and U. Obst-Sander and B. Wagner and M. Kansy and
D. W. Banner and F. Diederich},
journaltitle = {Org. Biomol. Chem.},
year = {2006},
volume ={4},
pages = {2364-2375},
title = {Multipolar Interactions in the {D} Pocket of Thrombin:
Large Differences Between Tricyclic Imide and Lactam Inhibitors}
}
\end{filecontents}
\documentclass{article}
\usepackage[style=chem-angew]{biblatex}
\bibliography{\jobname}
% Customisation
\DeclareBibliographyDriver{article}{%
\usebibmacro{bibindex}%
\usebibmacro{begentry}%
\usebibmacro{author/translator+others}%
\setunit{\labelnamepunct}\newblock
\printlist{language}%
\newunit\newblock
\usebibmacro{byauthor}%
\newunit\newblock
\usebibmacro{bytranslator+others}%
\newunit\newblock
\printfield{version}%
\newunit\newblock
\usebibmacro{journal+issuetitle}%
\newunit
\usebibmacro{byeditor+others}%
\newunit
\usebibmacro{note+pages}%
% Title inserted here
\setunit*{\adddot\space}%
\usebibmacro{title}%
\iftoggle{bbx:isbn}
{\printfield{issn}}
{}%
\newunit\newblock
\usebibmacro{doi+eprint+url}
\newunit\newblock
\usebibmacro{addendum+pubstate}%
\setunit{\bibpagerefpunct}\newblock
\usebibmacro{pageref}%
\newunit\newblock
\usebibmacro{related}%
\usebibmacro{finentry}%
}
\DeclareFieldFormat[article]{title}{#1}
\begin{document}
\nocite{*}
\printbibliography
\end{document}
I've also altered the formatting of the title so that there are no speech marks (as in your expected output in the question).