First Steps: Meeting your software needs
ChessBase Software
Unfortunately, it' s not as simple as opening your web browser and entering a URL, but it isn't too difficult. The first step is to get the right software, and there are two main options to choose from.
First, you can go to playchess.com and download the free client software. That will give you one month's free access to ChessBase's online server, which is where the show takes place.
A second option is to go shopping on the ChessBase website and purchase one of their playing programs (Fritz, Junior, Shredder, Hiarcs, Chess Tiger, etc.) for approximately 50 USD. The programs are all of appproximate world championship-level strength and have lots of neat bells & whistles, but the relevant matter at hand is that you receive two free ("free" given the initial purchase of the software, of course) 1-year memberships to the playchess.com server.
Windows Media Player 9 (or later)
You'll also need Windows Media 9 or later (Microsoft is up to version 10 now). It probably came with your machine if it's a recent one, but if not, you can download it here.
Part Two: I have the software; now what?
If you have the standalone client (option 1, above), then life is simple: execute the program and log on. If you have Fritz or one of its counterparts, execute the program, choose the Playchess.com option and log on. From this point, the two options will work the same way.
The next step is to navigate inside the server. Once inside, your interface will probably consist of four windows. One of them, probably on the bottom right, will have a header that says "Rooms." In that window you'll see a directory structure listing the server's rooms, obviously enough; the one you'll want to watch the lives shows is "Broadcasts." Click once and you'll be in.
The next thing to do is to go to the window on the upper left. It should have four tabs on the top (Info, Players, Games and World) - select the players tab. That will list all the players in the room, including yourself. My account name is "Initiative", and the easiest way to watch me give the show is to double-click on my handle while the show is going on. (More generally, since there are other people who occasionally broadcast shows, click on the "Status" bar and then double-click on the account name of anyone listed as a "Chairman".) And that's all there is to it! A new window will automatically open displaying the analysis board and providing the show's audio.
Great, but what about all those wonderful shows I've missed?
Part 1: Buying Ducats
Almost every past show is in the archives, but first you'll have to buy what ChessBase calls "ducats" (that's what they call their online currency). While you're logged on with a registered account, go to the Edit menu, select Payments and from there Fill Up Your Account. Follow the directions on purchasing ducats and you'll be good to go within 24 hours or so, possibly sooner.
CB charges about $13 US for 100 ducats, so given their charge of 2 ducats (about 26 cents) to watch an archived show (sometimes 1 ducat, sometimes 4 ducats, if the show was broken into two parts), it's unlikely that anyone with an internet account and the ability to purchase chess software is going to feel much of a pinch here.
Part 2: Accessing the Old Shows
Now that you're the proud owner of a bucket of ducats, here's what you do. Log on to the playchess server, look in the Rooms window and click on Chess Media System, then from within there continue to telescope in as follows: Radio ChessBase-->English-->Great Games with Dennis Monokroussos.
Once you've done that, click on the Games tab in the upper left (sub-) window and you'll see the list of available shows. When you find one you like, double-click and start watching! After a few seconds, you'll be asked if you'd like to watch the rest of the broadcast for the prescribed fee. If you accept, then that's it - all that's left is to watch and enjoy!