Library backup contains all your playlists. Just restore a backup onto the new PC.
Manually copy the backups across of course.
You will have to manually copy your Cover Art across as well. MC will rebuild Thumbnails based on that Cover Art once you have MC up and running.
Use the Rename, Move, and Copy Files function on the new PC to change all references in your Library from D:\ to E:\. There are lots of threads on doing this, plus the Wiki article on Moving Files and RM&CF function. Make sure to stop the automatic import in MC when you start it with an empty Library. See message in the bottom left corner. Also, turn off Auto Import in the old PC before you make a backup to transfer to the new PC. You can turn it back on after completing the transfer and changing the drive letter.
If you have special Themes or Custom Resources (keyboard shortcuts and such), you will need to manually move those as well. Also any third party plug-ins.
Best to keep the old PC working while you set up and test the new PC, and for a while after, so that you can check if anything important is missing. But the above are the basics.