By default the database is in C:\AlbumPlayerData. Copy of C:\AlbumPlayerData in full to the new computer.
In general when you put the albumplayer back in the same position and your music on the same file location, you can just install AP on your new PC.
Then after that you should start it. Upon start it will ask the database location, direct it to C:\AlbumPlayerData.
When your music is in same location as on your old computer, it should work again.
But if drive letter or folder changes of the music then you need to redirect the file locations so that the AlbumPlayer can find the files again.
You do that from the file menu in the collection editor.
There choose "redirect files to new location". See more info in the online help