Step 1 into getting help would be to give us the information we need to help you.

You say you encounter errors, but you do not specify in any way nor show the exact errors leaving us guessing.

Which OS/Distro is this and which version?

It should not be necessary to recompile aokex, however you appear to be missing the actual gmp library.

You keep saying that they tell you it's working fine. Once again, have you asked them whether they've compiled it with GMP instead of OpenSSL? I could never get it working with OpenSSL, GMP worked fine for me.

[root@***** ~]# yum install gmp gmp-devel
Setting up Install Process
Setting up repositories
update                    100% |=========================|  951 B    00:00
base                      100% |=========================| 1.1 kB    00:00
addons                    100% |=========================|  951 B    00:00
extras                    100% |=========================| 1.1 kB    00:00
Reading repository metadata in from local files
primary.xml.gz            100% |=========================|  78 kB    00:00
update    : ################################################## 227/227
Added 15 new packages, deleted 0 old in 0.71 seconds
primary.xml.gz            100% |=========================|  34 kB    00:00
extras    : ################################################## 131/131
Added 7 new packages, deleted 0 old in 0.20 seconds
Parsing package install arguments
Nothing to do

I am gonna get really noobie on you all..
Done that what next?

That user obviously does not have enough rights to install the newly compiled PHP module. You can try running "make install" through sudo, or use a root account. Or generally a user that has permission to write in /usr/local/lib/php/extensions/no-debug-non-zts-20020429/. You can check the permissions on that folder with "ls -l /usr/local/lib/php/extensions/no-debug-non-zts-20020429/", or, if you are using ACLs, with getfacl.
It would probably be unwise to change permissions of that folder, but you may check what groups it belongs to, and see if you can add the newly created user to that group.

OK, simpler then. If you have a root user, use it to do "make install". Do not use just any user, you need to do this with one that has sufficient permission. Root can always do just about everything, so, if you do not know how to give enough rights to another user, try running the command that gave you trouble with the root user.

Can't you directly change to that directory? And what do you mean, you only see these two files? That you are somehow locked into a directory and cannot get out? Try to enter "pwd" and see if it shows you the directory that you are in.
Also, what you "see" when you type "ls" or "dir" depends on certain setting for that user. Try a "ls -alF" on your root account for example, and you should see a lot more files. By default, a linux box does not show files that start with a dot for example.

Try to log your non-root user, do as you said until the "make" has finished, but before the "make install". Type "pwd" and write down the directory it shows. Log in the root user and do a "cd <directory you wrote down>". Unless your root user is in some kind of sandbox, this should directly get you into the right directory, so you only have to do the "make install" to get aokex installed.

Another thing you can try is log the non-root user, and try a "sudo make install". Depending on your configuration, you should be asked for your root password - the command is then executed with root-rights.
Or you can try to switch to the root user before "make install" by typing "su -". Again, you should be asked for the root-password, and afterwards be in the same directory, just as user root. do the "make install" then.