OK, thanks. We'll try the easy way first, and if that does not work out, there are other approaches as well.
I will assume that you have your AirPort Extreme functioning normally, but there is a setting that you will need to check and also a few settings to jot down because they may be needed when the AirPort Express is set up.
Leave the AirPort Express powered off for the moment and open AirPort Utility, select your AirPort Extreme if it is not already selected, and click Manual Setup at the bottom of the window.
Click the Wireless tab located just below the row of icons
Make sure that there is a check mark in the box next to "Allow this network to be extended"
Jot down the name of the wireless network and also the exact setting that you see for Wireless Security. I assume that you know your wireless network password
Click Update if you made any changes and allow the AirPort Extreme to restart, which will take 25-30 seconds.
Next, we want to perform a Factory Default Reset on the AirPort Express to get it back to default settings so that it can be configured again. Move the Express close to the AirPort Extreme and your computer.
Hold in the small reset button on the Express (next to the audio port) with an unfolded paperclip or something similar and keep holding it in while you plug the Express back in to power.
Keep holding another 10-12 seconds after you plug it back in and then release the reset button
Allow 40-45 seconds for the Express to restart to a slow, blinking amber light.
If you have a spare Ethernet cable handy, temporarily connect your computer and the Express together. If you don't have an Ethernet cable, we will use wireless to configure the Express.
Open AirPort Utility and the AirPort Express should appear. If you use wireless to connect, you may be asked to "Switch" networks, that's OK, allow the switch to occur.
Click Continue to follow the guided setup in AirPort Utility
Step 1 will ask you to assign a name and device password. Write the password down for future reference. You can edit the name as you wish if you do not want to use the name that AirPort Utility suggests
Step 2 click the "join my current network" option
Step 3 click the "Join" option again
Step 4 click the small arrow on the selection box next to wireless network name and the name of the AirPort Extreme network should appear. Click it to select it.
Make sure that Wireless Security is the same setting that you wrote down when you checked the AirPort Extreme setting. Enter you wireless network password and confirm.
Important.....Be sure to enter a check mark in the box next to "Extend the range of this network"
Step 5 click Update. As oon as you see the message that the settings have been stored successfully, close AirPort Utility and disconnect the Ethernet cable if you used it.
Wait 40-45 seconds for the Express to restart. If the settings are correct, you will see a green light.
Now, move the Express to a location that is approximately half the distance from the AirPort Extreme and the area that needs more wireless coverage and let it restart. Be sure that you log back on to your AirPort Extreme network with your computer and you should be in business.
Please post on your results.