How can I empty the trash with old backups in it? Since my backup disk is full, I wanted to delete old backups to free space. Since I didn't figure out how to do it on TimeMachine itself properly, I selected all the folders with backups on the TimeMachine disk and put them in the trash.

Now, however, I can't empty the trash/ delete those items. Whenever I try, it shows that "Library is in Use", even when I close all tabs and applications.

How can I empty the trash and really delete those old backups so the space on my backup disk is free?
Okay, before I get into more detail, let me quickly tell you this:

You should not delete TimeMachine backups yourself like this!

The filesystem on a HD formatted for TimeMachine works differently from what you're normally used to. Without going into too much detail, the main difference is that it uses hard links. That means that files that look like they're a simple copy, are actually just links, pretty much like an Alias that you can create in the Finder. That's how you can see each backup folder as a full mirror of your system (or all files that are backed up) without really having x copies of them (where x is the number of backups taken). This saves disk space (by a lot).

Instead, you should simply let TimeMachine sort this out on its own. Once it runs out of disk space, it automatically starts removing older backups.

Btw, for the same reason you probably don't want to store anything else on the TimeMachine partition. It's simply not a "normal" disk, from the perspective of the average user.

To actually answer: There are several ways to force delete from/clear the trash bin. The first one I'd try is to open it, right-click the problematic file(s) and chose "Delete immediately..." Other methods usually involve Terminal and require some shell knowledge. A quick google search revealed this, which also covers your system version it seems. In short: running 

sudo rm -rf /Volumes/YOURTMDRIVE/.Trashes/*

in Terminal should work, but again, I think you shouldn't go down that path anyways. (YOURTMDRIVE would be the name of your TimeMachine volume, watch out for spaces, those must be escaped with a preceding \)

(That being said I don't think you're running risk of ruining anything fatally, but TimeMachine will now probably take a while to fix the backups when it runs the next time).

Last note: You can read up a bit about how TimeMachine works here