Yes, connecting a 1080p TV to a modern PC via the HDMI connection will work.
For best results set the TV to
1:1 pixel mapping mode, depending on the TV make this may be called something else, like "Just Scan", "dot by dot" or "PC mode". And turn picture features like noise reduction and edge enhancer off - as they will make the PC image look poor.
Most graphics card (including the HD 7850) have a native HDMI port - so setting it up should be pretty simple. Though you may need to turn off overscan using the catalyst control panel if the image doesn't fit the screen or runs over it.
As I said, TVs will work with a PC - however they are not usually ideal monitors. This is because the pixels are much bigger than on a PC monitor, so if you are sitting relatively close the image may not look too sharp. Also, TVs (due to the image processing they do) tend to have much greater input lag than PC monitors - though the extent of this issue varies widely between TV models.