The AirPort Extreme cannot wirelessly extend the signal from a non-Apple router.
If you cannot install an Ethernet cable to connect the existing router and AirPort Extreme together, you might consider taking a look at a pair of Ethernet powerline adapters to send the Ethernet signal over the AC powerlines in your home.
The theory with these devices is that you already have the wiring in place....you just need the adapters.
In practice, things do not always work as well as the theory would suggest, so if you decide to try a pair of these devices, it might be a good idea to understand the store's return policies before you buy.
Any computer / electronic superstore will have a selection of Ethernet powerline products from which to choose. Ask one of the sales guys what might work in your home.