This is what I do when I have a very similar problem (but mine is Linux via Linux and I use port 5901 for VNC):

First, we make it so that all connections to localhost:13389 on your laptop will go to the intermediate server (on port 3389):

laptop$ ssh -L 13389:localhost:3389 my_user@LINUX_SERVER_IP

Then, we make it so it that all connection to localhost:3389 on the intermediate server are forwarded to the PC behind the firewall (on port 3389):

my_user@LINUX_SERVER_IP$ ssh -L 3389:localhost:3389 'PRIVATE_DOMAIN\my_user'@PC_NAME

(note that this command is run inside the interactive shell on the intermediate server.)

Now, you should be able to connect to localhost:13389 and access port 3389 on the remote PC.

Debugging

Since it isn't working, there's a few things we can try. We'll do in a way to isolate where the issue is:

1. On the remote PC you want to access, can you telnet localhost 3389 to ensure it's open and ready for connections? Microsoft has a nice article on it
2. If that works, can you try to execute telnet localhost 3389 on the intermediate server to check it's forwarding correctly to the remote PC?
3. Finally, telnet localhost 13389 on your laptop, to see if it's forwarding all the way through.

As soon as you hit an error stop there and please add a comment so we figure it out.

I ran into the same black screen + disconnect problem myself today, using putty as my client. I did find a solution eventually.

I switched from putty to bitvise tunnelier, and setup a S2C connection with the following settings:

listen if:0.0.0.0
listen port:13389
destination host:localhost
dest port:3389

As chance would have it, I'm using bitvise ssh server on my server, so this may just be a happy combination for two products made by the same vendor. It would be great if this solves the problems for others.

For the record, I'm not affiliated with these guys in any way.