I live in an apartment where I am allowed to use the wireless router's connection for the internet but the speed is sometimes very shaky and weak. Is is possible to use a D-Link DI-524 wireless router that I have as a hub to connect to the wireless connection I am already using in order to increase the speed and stability of my internet connection?
It sounds like you're looking for a "wireless client mode" feature that will let your router connect to the other wireless network as though it's just a plain wireless adapter. A lot of routers don't include this feature in their stock firmware, which is a shame. It's hard to say just from looking at the specs of that router, and the little bit of Googling doesn't reveal much. I would need to look in the firmware's web interface to be positive, but my guess is, no.
You didn't ask this, but as far as other devices that will allow you to do this, I personally own the Asus RT-N12/B which has an awesome switch on the back to let you change modes between router, repeater, and access point. I use it as a repeater with my main router to provide stronger signal on another level of our house and it also lets you plug wired devices in. All you do to set it up is put in the SSID and PSK of the main wireless network.
I know other devices have this feature, that's just the one I use and know, and tends to be a pretty good bargain.