You might want to check the mysql log. On my machine it's located at /var/log/mysqld.log. You could try something like this:
or you could run this in one window
tail -f /var/log/mysqld.log
and then try to start mysql in the other.
if the file /var/run/mysqld/mysqld.pid doesn't exist, then mysql was probably not able to start up because it could not create that file. It could be a permissions problem or something? I'm not really sure.