well first of radio frequency of wifi isnt realy ideal for long range with no clear line of sight and being in major city etc...
first you want the best antenna setup for your router to begin.so i suggest you have a router with removable antennas or else
its gonna be realy realy hard to.like this router for example
http://www.newegg.com/Product/Product.aspx?Item=N82E16833704038 ,it comes with 3 antennas that you can remove and repalce with stronger rp-sma antennas,the ones most routers come with omnidirectional so its harder to reach a further location.what you then get is rp-sma 'pigtail' wires so you can plug one into a rp-sma connector to router and the antenna you want to use.make sure you mount router near a window in direction you will be giving the wifi to.you can use a satellite dish too,http://www.randallcole.com/dish/, he also used a 'cantenna' he made out of a can with an appropriate
diameter and height usually there about 3.25 inches in diameter,this site tells about the cantenna, http://www.turnpoint.net/wireless/cantennahowto.html, in the calculator it has 3.25 for normal cantenna diameter, and the 1/4 guide wavelength is about 2.49 so i would make a hole for a n type connector from the bottom of the can to the top.watch how to make cantenna,http://www.youtube.com/watch?v=NVjN9EK7B0E,make sure to get solid wire of copper wire not stranded!
you could also use a biquad antenna which you can also make but alot easier and cheaper buying on ebay for about 20 dollars,
http://www.ebay.com/itm/WiFi-Antenna-Biquad-Cantenna-Dish-Wireless-Booster-Long-Range-GET-FREE-INTERNET-/130957136570,
dont forget the rp-sma pigtail for it too! you can also mount the biquad to a satellite dish which is realy good.
you can try buying a parabolic antenna too,http://www.ebay.com/itm/Parabolic-Antenna-2-4GHz-24dBi-WiFi-Grid-Antenna-Directional-Outdoor-Wireless-/190924015275?pt=US_Directional_Network_Antennas&hash=item2c73f546ab,.
and now for the computer that is trying to get the wifi personally i would get a decent usb wifi adapter with external antenna(rp-sma)
such as tplink 722n,http://www.ebay.com/itm/FREE-SHIP-TP-Link-TL-WN722N-150Mbps-High-Gain-Wireless-USB-Adapter-/221187427240?pt=LH_DefaultDomain_0&hash=item337fcc4ba8,i would use a biquad antenna as they give you great gain and for better results i would waterproof the biquad antenna by putting some tupperware on the biquad and put the dish with the adapter and biquad outside and point in the direction your trying to get from router,make sure to waterproof the adapter and wires! you could also put cantenna on the dish but its easier in my eyes to use biquad.try not to get cheapy wifi adapters too.