You could do it if you got a wap that is capable of acting as a gateway out Ethernet attached directly to the core router on a separate network and separate wireless id name than the main router (separate from the core only connecting to the far router over wireless)
you may be able to do it with fairly low budget pre made wireless access points, but dont take my word for it do some checking beforehand.
you will definately be able to do it with a pc of some sort with a wireless card (put nice antennas on it since it will have to go the long haul) (id strongly recommend using Linux for this computer look up Linux distros intended for routing). look to spend around 100 bucks plus a wireless card
http://www.frys.com/product/6214480
70 bucks triple core amd with ati graphics and mobo, all u need is the cheapest 9 dollar ram u can get a case a power supply and a thumbdrive to hold the os.
can be done cheaper than that example but that is a powerhouse for 70 dollars