Xe(La)TeX uses Computer Modern as default typeface.
Computer Modern does not contain real accented characters, they can be composed of several characters via macros instead.
Consequently you will note that using \'o in the document directly will produce "ó", but writing ó doesn't cut it.
By default Biber converts these macros to their corresponding Unicode characters and sends those to the .bbl file.
So effectively we are requesting ó and not \'o in the document even though you wrote \'o in the .bib file.
You can make Biber output the combining macros instead of the Unicode characters by compiling with the --output-safechars command line option.
I would, however, advocate a different approach. If you load fontspec
\usepackage{fontspec}
the default font is set to Latin Modern which contains lots of accented characters natively.
So we can just go on using ó in the document and it will appear in the final PDF.
MWE
\documentclass{article}
\usepackage{fontspec}
\usepackage[
backend=biber,
style=alphabetic,
citestyle=authoryear
]{biblatex}
\usepackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@article{Beck_Johnson_2013,
title ={The Effect of Temperature on \textit{{Anopheles}} Mosquito Population},
author ={Bj{\'o}rnstad, Ottar N.},
year ={2013},
journaltitle ={PLOS ONE}
}
\end{filecontents*}
\addbibresource{\jobname.bib}
\begin{document}
\cite{Beck_Johnson_2013}
\printbibliography
\end{document}

author = {Bjórnstad, Ottar N.},. – moewe Dec 02 '15 at 07:40\', but not for\v, near as I can tell. – SEC Apr 09 '16 at 14:31author ={{\v{C}}j{\'o}rnstad, Ottar N.},in the MWE above. You font will have to contain the requested characters though. – moewe Apr 09 '16 at 14:48fontspecand compile withxelatex, but not if i loadnewtxtextin lieu offontspec, which is what i tend to do. hm. – SEC Apr 09 '16 at 14:52fontspecor use Biber with the--output-safechars. The whole point was that XeLaTeX on its own doesn't cut it. – moewe Apr 09 '16 at 14:53newtxtextcouldn't display aČif it was directly input as a UTF-8 character, only the composite\v{C}worked. That makes me think the font doesn't have the characters available natively and needs to use the composite version. You can get that from BIber with--output-safechars. – moewe Apr 09 '16 at 15:01