When I upload my profile photo to Facebook created in cs6, then it looks very cheap in quality.
After uploading the profile picture to my Facebook Page
The image you are showing us doesn't look like an original screen capture but interpolated (scaled up) image of a screen capture. Because even the other UI elements from your computer included in the screen capture are interpolated, it is obvious that your facebook page is not displayed the way you are showing it to us.
Regarding image resolution, you don't need to know any resolution and ppi for displaying images on screen (web). All you have to do is view your image at 100% zoom in Photoshop to know how your image will appear on screens if it is not scaled. If you have an image that will be reduced in size either by you or a website when used on the web, use File > Save for Web, reduce the image size there and refer to the preview result for the final appearance. However, usually websites that change the size of posted images reduce them, while your problem seems to be caused by increasing the size from a smaller image than your original which is worse than reducing size.
DPI (or PPI) doesn't matter on the web - look for pixel dimensions, X by Y. If you save imagery as .jpg its all @72 ppi. Screen sizes viewers use on the web are all over the place now if you consider smart phones and hi res displays. Start with some typical sizes (800x600, 1280x1024 etc.) or make some up! I don't do FB but designing for that isn't any different than much else. You make things 'big enough' to look good but not much bigger Experiment!