Increasing shutter speed of Nikon D3200 while using the built-in flash or a speed-light?

I'm very new to DSLR photography having recently bought a Nikon D3200. I was trying increase the shutter speed in an evening shoot so that I could darken the background while still illuminating my subject ahead using my built-in flash. I then realized that I could not increase the shutter speed beyond 1/200th of a sec.

My query is that it possible to increase the shutter speed of my camera using speed light specially the entry level ones like Yongnuo YN 560-II or Nikon SB-400 or I need to go for more higher end version.

No. The camera's sync speed is 1/200 and that's as fast as the shutter will go using flash.

It doesn't matter what version of speed light you have, 1/200 is all you get.

1/200 is the fastest shutter speed you can use with that camera when using any flash.
Reason is that that is the fastest speed at which the shutter blind is fully open. Any faster than that and exposure is achieved with a "slit" that travels across the sensor. As the flash duration is as quick as 1/10000th sec you would get only part of the frame exposed.

Canon cameras and high range Nikons have high speed sync that allows flash up to 1/4000th with external flash, but unfortunately your camera doesn't.

With some higher flashes like the SB700 and some higher bodies like the D7000 there's an FP mode that allows the flash to go through a continuous burn through a shutter speed faster than x-sync. I have tried this using my D90 and SB600 which has this feature. It is not worthwhile, as the flash is less intense in that mode.

In case you are wondering FP stands for focal plane, and comes from focal plane flash bulbs that used to be available to work with film SLR's. The FP sync would trigger the bulb just before the shutter starts to open, and the bulbs were designed to burn with an even intensity during the 1/60th sec while the slit of the focal plane shutter (that was set faster than 1/60th second) moved across the film plane.