Requirements according to the site:
* PHP >= 4.3.0
* PEAR >= 1.4.0

Automatic compile and install using yum and pecl

Make sure you have /usr/bin/pecl:
# yum install php-pear

Make sure you have a C++ compiler:
# yum install gcc-c++

Make sure you have /usr/sbin/apxs:
# yum install httpd-devel

Note sure, is re2c needed? Pecl put up a warning when it was not available:
# yum install re2c

PECL does the rest:
# pecl install APC

Skip to 'Using APC' below for configuration.

Manual installation (untested)

This uses the phpize command from php-devel.

Download the latest version:
# wget http://pecl.php.net/get/APC

Download a specific version, untar, make and install:
$ wget http://pecl.php.net/get/APC-3.0.19.tgz
$ tar zxvf APC-3.0.19.tgz
$ cd APC-3.0.19
$ ./configure
$ make
# make install

Using APC

Load the extension:
# echo "extension=apc.so" > /etc/php.d/apc.ini

Runtime configuration (use all defaults):
# echo "# See http://www.php.net/manual/en/apc.configuration.php for APC configuration directives." > /etc/httpd/conf.d/apc.conf

Reload Apache:
# service httpd reload