I have 1 computer that has both wireless and wired network cards, 3 that only have ethernet, and 1 4-port router.

2 are running Linux Mint 13 & 1 is still being configured as a home server. All are Desktop computers.

on my wired network I don't have any internet connection but I need to allow all to access the internet. My wired network is on 1 ip addressing while my wireless is on another.

here is where it gets alittle complicated....I am connecting to the internet via my neighbors wireless while still needing to keep my LAN up and accessible.

How can I easily share out my wireless over ethernet to give internet access to my whole network?

all are DHCP addressing.

Does your neighbor know that you are using his internet?

yes he does know because it is locked down to where I have to know his wep key in order to access.

I think you should look into using your internet connected computer as a router.

I've never done that, but the Arch wiki is usually a good reference. You might start there: https://wiki.archlinux.org/index.php/Router