First do check you are not sitting the same channel (or indeed a close one) with your neighbours- quite often I see people pile on 1 and 6 and leave everything else up (channels go up to 12 at least- more and it starts getting odd), make sure you do not have the "security" option low power transmission active and if you have a proper antenna on the thing try tweaking it (even a basic antenna is not directional but direction does matter- you could learn radio engineering/antenna design* but that is a very specialist field and frankly messing around with
http://www.passmark....lessmonitor.htm and shouting up and down stairs does about as well). Equally it will see diminishing returns as the years drag on but wireless N can work (do check and make sure) in the 5MHz range which was previously 802.11A (mainly commercial use) although that changes things again.
*I can not recall which at this point but one of the big hacker conferences (American I think) had a well nice crash course on it that I saw.
Anyway that over I know you said you wanted wired connection to the second access point which has probably said most of what you want to know (get any old router and stick it on there- it might take some fiddling depending upon how you want to set things up though) but I would look to wireless instead.
You can redo an old PC to turn it into a router and access point (and firewall, network share server, other servers.....) but right I reckon the cheapest method right now is
Shake down ebay/amazon/chosen online tat merchant/a couple of mates* for a BT Homehub v2.0B (could be a 2.0A or 1.5 but those are harder- 2.0b can be done with a USB drive, a copy of windows and a wired network cable). The are going for next to nothing right now but that will probably change in the coming months.
*BT give these things out like sweets and most end up sitting in cupboards as their stock setup is awful or replaced with newer models.
Flash it with a custom firmware (
http://gbatemp.net/t...ome-discussion/ )
Depending upon your router you have various methods to extend the range from methods like WDS (nice but requires a given set of conditions) to various sorts of bridging (
http://wiki.openwrt....owto/clientmode ).
I have done such silly things as run a massive ADSL (RJ12) cable or repurpose a network cable (you might see phones in offices run on network cables- same idea) to stick routers up high when phone sockets are down low and resiting the point is not an option but those are not ideal.