Q: Cannot access internet on strong wifi

Restart your router and modem.  Check your router settings Recommended settings for Wi-Fi routers and access points - Apple Support  Contact your ISP to make sure everything's fine on their end.  If so, contact the router manufacturer.

Try creating a new user on your Mac and test it there.

It can certainly be an issue specific to this Mac, but the router can be blocking just this mac for some reason.  If the issue is in both users, I would honestly press and hold that reset pinhole on the back of most routers before continuing only because your Mac shows its fully connected, but its just not getting the details it needs from the router.  If you were having an issue connecting to the router period, then it could possibly be an issue with the mac or some configuration conflicts.  I hope that helps.

So your Mac is connecting to the router and getting the thumbs up but the DHCP service on the router is not giving your Mac an internal IP address.  I can tell that because you are seeing the self-assigned IP address of 169.254.X.X  This occurs when the router has failed to deliver an IP address, the Mac, and all computers assign that address to itself until an IP is delivered.  At this point I can tell you for sure that it's not your Mac itself, but something in your network.  The router is what assigns the internal IP addresses to all the internal devices using a service called DHCP.  There are a number of reasons this service can fail to assign an address.  it could be blocking the connection or it could have reached a device limit.  I would suggest contacting your router manufactuer at this point.

Have you tried creating a New Network Location on the iMac? It can sometimes help solve wifi connectivity problems:

Using network locations (Mac OS X v10.6 and later) - Apple Support

Also, here are a few other wifi troubleshooting tips that have helped some:

http://osxdaily.com/2012/11/30/resolving-stubborn-wi-fi-connection-problems-in-m ac-os-x/

If you have a Mac wiz you can call upon it's worth giving them a call.

The first link I posted for creating a new Network Location is quite straightforward - and you can always easily return it to its current 'Automatic' setting if it has no effect.

The process is:

• From the Apple Menu (top left of the screen) select System Preferences > Network.
• Select WiFi on the left and then 'Locations'. From the drop-down menu and choose 'Edit Locations'.
• Click on the '+' icon to add a New Location and give it a name > click on 'Done' and Apply the changes.
• The newly created location should now be selected in the drop down menu.
• With Wifi selected on the left give the Mac a few moments to recognise the new location and hopefully the wifi connection will show as 'green'. If it doesn't try switching wifi off and then back on again and hopefully it will go green this time around. Once green, test to see if you are connected.

If, after all this, there is no change you can return to the 'Automatic' setting that was the previously selected option in Network > WiFi.

Start up in Recovery mode. In the OS X Utilities screen, select Get Help Online. A clean copy of Safari will launch. No plugins, such as Flash, will be available. While in Recovery, you'll have no access to your saved bookmarks or passwords, so make a note of those before you begin, if they're needed for the test.

Test. After testing, restart as usual and post the results.

I'm not sure whether you actually tried to join the Wi-Fi network while in Recovery. That was the idea of the test.

Please connect to the router with an Ethernet cable. Does that work?