APC can be installed as a package on Amazon Linux AMIs.
It's enough to just add it to the package to be installed in one of your
.config file in
# .ebextensions/myapp.config packages: yum: php55-pecl-apc: 
See Amazon's documentation for more info.
Logs of the execution of
.ebextension/*.config files are found in
You should see something like
[DEBUG] Installing/updating [u'php55-pecl-apc'] via yum [INFO] Yum installed [u'php55-pecl-apc']
To search for APC packages:
yum search node