I actually researched the subject of transparency on the Windows clipboard extensively, so while I'm not really experienced with javascript, I can help you along with the principles.
The clipboard works with a system of string IDs for their clipboard types, and you can put multiple such types on the clipboard simultaneously. In recent years, a lot of applications (including MS Office and Gimp) have started using the type "PNG" for transparent images, which will contain a raw byte stream with the bytes for a png image.
However, a much more commonly used one is the DIB format, which has clipboard ID "DeviceIndependentBitmap", and that one may be... problematic. Like the PNG one, it is a raw byte stream, but the actual file format is a bit of a mess. For more information about the DIB format, I advise you to read through this question and the answer I gave to it, and the Device Independent Bitmap specs and BITMAPINFOHEADER struct on MSDN. Long story short, it's a 32bpp RGB format that's abused as (and sometimes mistaken for) ARGB. I have no idea how plausible it is to parse DIB into a usable image in Javascript, though, but a lot of programs (including Google Chrome) seem to use it as only transparent image format they put on the clipboard when copying an image.
To fully support transparent pasting you'll probably need to support both the PNG and DIB formats (and, given the problems and controversies surrounding DIB, preferably in that order). This answer may give more information regarding the general method to sift through the clipboard and parse the PNG and DIB images, though it is C# code, and not Javascript.