I haven't finished reading all of MusicHawk's replies, but I have a few minutes and I thought I'd reply directly to you, Chaoscar...
Yes, you will want to add the MP3 files to the laptop first. Preferably, you will want to move these over to the same "path" as where they were located on the original machine, though this looks like this isn't possible due to your partition issues. I'll get there in a second.
When I say "transfer the library" I am NOT referring to the MP3 files themselves. In MC's terminology, the "Library" is the database that MC uses to display, sort, and search all of your music files. It contains all of the tag data that can't be embedded directly in the files themselves. It is a set of relatively small files, mostly of the type JMD, and is specific to JRiver's MC. By default, the library is stored here:
Windows Vista/7: C:\Users\<USERNAME>\AppData\Roaming\J River\Media Center 14\Library
Windows XP: C:\Documents and Settings\<USERNAME>\Application Data\J River\Media Center 14\Library
However, you can have multiple libraries and libraries in non-default locations in MC, which are defined via the Library Manager tool found via File -> Library -> Library Manager. Assuming you have NOT changed from the default library, then you will want to copy the this folder (the Library folder listed at the path above) over to the new laptop. If you are going from XP to Windows 7, you will copy it from the XP location over to the new Windows 7 location. There's no problem with this, you just have to put it in the right place.
You can also accomplish this same goal using the Library backup and restore tool. If you don't know what you are doing, this might be easier. On the original machine, simply go to File -> Library -> Back up Library and save the ZIP file somewhere you can find it (like on your desktop). Copy this ZIP file over to the new computer. Then, in that copy of MC, go to File -> Library -> Restore Library and find that ZIP file you copied over and choose it.
You also will need to copy over the media files themselves. Since you can't put them on the new computer on the same drive letter, you will have to do some maintenance on the Library when you are done. This shouldn't be a huge deal, really. The most important thing is to keep them in the same "relative" organization and location. So, if you have all of your media files currently at: M:\music\, then the easiest thing do do would be to copy the entire "music" folder over to the new computer and put it at C:\music\. Just make sure you keep all of the subfolder structure and file names "underneath" the "music" folder the same when you copy it over (don't re-organize your music folder while you transfer it over). Then, fixing MC to look for C:\music\blah\blah.mp3 won't be a big deal at all. If you describe in detail exactly where your files are now, and where you want to move them, I can explain this further.
So, here are some steps for one way to do it:
1. Transfer all of the MP3s over to the new computer.
2. Install MC.
3. Copy the Library folder over to the new computer.
4. Do the "registry" transfer I explained above.
5. Open MC and fix the library to point at the new files.
For step 5, let's assume that you are doing it as I described above. Say that originally, your music was all found at M:\music\ with maybe Artist and Album subdirectories or something. So, one of your original music files might be found at "M:\music\Pink Floyd\Dark Side of the Moon\2 - On The Run.mp3" or something like that. In this case, the easiest thing to do would be to move the whole \music\ folder over to your new computer and keep it on drive M. However, you might not be able to do that, so you might need to move it to C:\music\ or something like that. That's okay, as long as you keep the \Pink Floyd\Dark Side of the Moon\2 - On The Run.mp3 part of each filename the same.
Even if you want to move it from M:\music\ on the original computer over to C:\Users\<USERNAME\music\ on the laptop, that's fine, so long as the files end up being C:\Users\<USERNAME\music\Pink Floyd\Dark Side of the Moon\2 - On The Run.mp3 in the end.
Then, once you are all done, open MC up. You'll notice that everything looks the same as it did on the original computer, except that nothing will actually PLAY. All of the files are "broken" and will get little red Xs over the icons. You need to fix this. This is simple using the Find & Replace tool. Simply select all of the files in MC, and right-click and choose Library Tools -> Find and Replace. Then, in the Find And Replace dialog box, check ONLY the Filename tag box in the list on the Left. Then in the Find what box, you want to type in what part of the filename you want to replace and in the Replace box, what you want to replace it with. So, if you need to turn:
Original: M:\music\Pink Floyd\Dark Side of the Moon\2 - On The Run.mp3
New: C:\music\Pink Floyd\Dark Side of the Moon\2 - On The Run.mp3
It is simple. Just type "M:\" in the Find What box and "C:\" in the Replace box, and hit go.
If you need to turn:
Original: M:\music\Pink Floyd\Dark Side of the Moon\2 - On The Run.mp3
New: C:\Users\chaoscar\music\Pink Floyd\Dark Side of the Moon\2 - On The Run.mp3
Then that is still easy, just put "M:\" in the Find What box, and "C:\Users\chaoscar\" into the Replace box, and hit go. Once you do this, MC will go through and change the filename (path) for EVERY file that you've selected to the new location, and "fix" them.