How to extend range of a wireless router with a second wireless router

Consider this scenario. You have a modem router in the second floor of the house, whose range doesnt extend to the first floor. You have access to a low cost wireless router and would like to use it to extend the range of the router.

First, note down the wireless settings on the first router. Eg. My primary router is a Belkin N150 Wireless Modem Router F9J1001. It can configured on 192.168.2.1.

With Wifi normally connected, First login to the primary router’s settings page, and take note of the following wireless settings:
Wireless > Channel and SSID:
Wireless Channel: 11
SSID > belkin.31e
Wireless Mode > 1×1 802.11 n
Bandwidth > 20 Mhz

If any of these settings is set to auto, you need to change it to one of the other modes.

Now, access the second router’s configuration page. For me, it is a TP-LINK TL-WR740N Ver 4.27. I decided to change the stock firmware on the router to dd-wrt since the stock firmware was proving troublesome. You can try to find out where to access the same settings on your own router.

In Win 8, Control Panel > Network and internet > Network connections:
Right click on your Wifi connection and Disable it.
Enable your Ethernet connection, and connect the secondary router to your PC with an Ethernet cable.
Now Login to 192.168.0.1 (the default IP for a TP-LINK) or 192.168.1.1 (the default IP for dd-wrt) and set up the secondary router.

For dd-wrt, the settings I modified are:
Setup > Basic setup > Network setup > Router IP:
Local IP address: Change from 192.168.0.1 (default) to 192.168.2.6
Gateway: Set as your Primary router’s IP, i.e 192.168.2.1
i.e if your Primary router’s IP is 192.168.A.B, change the secondary router’s IP to 192.168.A.E leaving some amount of digits for other devices.
Assign WAN Port to Switch: Dont modify (unchecked)

Save.

Wireless > Basic Settings
Wireless Mode: Client Bridge (Routed)
Default GW Mode: Manual [Even Automatic DHCP seemed to work at least initially]
Gateway: Set as your Primary router’s IP, i.e 192.168.2.1 [If you skip this step, your secondary router may fail to renew IPs after getting disconnected once)
Wireless Network Mode: N only
Channel Width: Full (20 Mhz)
Wireless Network Name (SSID): belkin.31e (Set it to the same SSID of the primary router)

Wireless > Virtual Interfaces>
Add:
Wireless Mode: AP
Wireless Network Name (SSID): dd-wrt_vap (Autogenerated string)
Wireless SSID Broadcast: Enable
Save and Apply.

Now disconnect Ethernet, and Enable Wifi adaptor. Now you should be able to browse the internet after connecting to the secondary router via Wifi.
On dd-wrt firmware, the whole process was very straightforward.

Reference:
TP-link repeater bridge wiki (Note that there is no “repeater bridge” mode in newer firmware).
Client bridge dd-wrt Wiki