How do I get "et al." to appear in italics when using \textcite or \citeauthor with biblatex

"et al." corresponds to the localization key andothers. In most styles it is set by the generic bibliography macro name:andothers. You can redefine this macro to wrap \bibstring{andothers} in \emph.

\documentclass{article}
\usepackage[american]{babel}
\usepackage{csquotes}
\usepackage[style=chem-rsc]{biblatex}

\renewbibmacro*{name:andothers}{% Based on name:andothers from biblatex.def
  \ifboolexpr{
    test {\ifnumequal{\value{listcount}}{\value{liststop}}}
    and
    test \ifmorenames
  }
    {\ifnumgreater{\value{liststop}}{1}
       {\finalandcomma}
       {}%
     \andothersdelim\bibstring[\emph]{andothers}}
    {}}

\addbibresource{biblatex-examples.bib}

\begin{document}
\textcite{companion} \citeauthor{companion}
\printbibliography
\end{document}

This is another case where egreg's xpatch package comes in handy. Instead of copy-pasting the whole content of the name:andothers bibmacro, one may selectively apply changes (as suggested by Marco Daniel):

\documentclass{article}

\usepackage[american]{babel}
\usepackage{csquotes}
\usepackage[style=chem-rsc]{biblatex}

\usepackage{xpatch}

\xpatchbibmacro{name:andothers}{%
  \bibstring{andothers}%
}{%
  \bibstring[\emph]{andothers}%
}{}{}

\addbibresource{biblatex-examples.bib}

\begin{document}
\textcite{companion} \citeauthor{companion}
\printbibliography
\end{document}

Another way to achieve this is as follows:

\usepackage[maxnames=6,minnames=3]{biblatex}
\DefineBibliographyStrings{english}{%
    andothers = {\em et\addabbrvspace al\adddot}
}

You actually have control over the whole text and its styling, and you can control for different languages.