Depending what AirPort Express model you have, it is either possible or not.
If you have the first generation (2008 - 2012), then it is not possible, since it only has one Ethernet port. The 2012 model has two Ethernet ports (one for the connection from the modem to the AirPort Express and one for an additional device with Ethernet, like a switch or router.
Anyway, it is probably best to get a switch, if you have the 2012 (current) model. If not, it will probably be more tricker and a router needs to get involved.
3) Connect a cable from the router to the Express, letting the express create the WiFi Network. You will need to experiment with the settings in the Express, probably setting it to extend a network
// Gunnar
You don't want to tell the express to extend the network. You want to tell it to create a wireless network. What is key, is that the wired router you buy should be handling your NAT and handing out IP addy's via DHCP. Your express should be set as a wireless access point in bridge mode. You can google how to do this so I'm not going to waste internets explaining this here...
Good luck!
Would this still allow me to share stuff from a wired computer to one connected via wifi?
I'm thinking its probably going to be best for me to get an extreme.
Does anybody know the answer to this?