Update 26-Oct-2020
I now recommend Radmin VPN instead of Hamachi. I've had far less issues with this software than Hamachi.
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In my opinion, Hamachi is currently the best VPN software for creating a virtual Local Area Network environment for gaming. We've lost some great alternatives over the years, but Hamachi has been around forever and still works well. The application is easy to setup and use, but unless you're very lucky, you're probably going to run into problems along the way.

I've been using this program for several years and have had my share of troubleshooting mishaps. In that time, I've scoured the internet for solutions to connection problems and have found that the following steps have gotten me the most success. This is a quick start-to-finish tutorial for dummies on setting up Hamachi and ensuring it actually works!

Note: If you're playing vanilla multiplayer Diablo and you want to skip the headache, just use GameRanger. It should work right out of the box. However, if you're playing with a mod like "The Hell", GameRanger won't work and you WILL need something like Hamachi. Plus Hamachi is just a tidier program with no ads to hassle with.

Each person on your network needs to follow these steps:

Windows 10

1. Download, install and run LogMeIn Hamachi from www.vpn.net.
2. Install your game to a folder on your C Drive. (NOT to Program Files. Example: "C:\Game Files\Diablo")
3. In your Windows 10 Search box, search for and run "allow an app through windows firewall".
A. Click the button, "Change settings"
B. Scroll down until you've found "Hamachi Application" and check the "Private" and "Public" checkboxes.
C. Click "Allow another app", browse to your Hamachi install folder and select, "LMIGuardianSvc" to add it to your firewall list. Check both "Private" and "Public" check boxes.
D. Look for your game in the list. If there, do the same as above. If not, you'll need to click, "Allow another app", browse to your game folder and add ALL executables that pertain to the game itself (This may not be necessary, but let's be safe rather than sorry). Remember to check both "Private" and "Public" check boxes.
E. Click "OK" and exit the window.
4. In your Windows 10 Search box, search for, "Control Panel", then when the window comes up, search for and open, "Network and Sharing Center".
5. In the Network and Sharing Center window, click "Change advanced sharing settings".
A. Expand the arrow to the right of "Private"
B. Check the box, "Turn on network discovery", and "Turn on file and print sharing".
C. Expand the arrow to the right of "Guest or Public"
D. Check the box, "Turn on network discovery", and "Turn on file and print sharing".
6. Open up the window, "Network and Sharing Center" again as you did above, then click, "Change adapter settings".
A. Right-click on the icon labeled, "Hamachi" and select "Properties".
B. Click on the line next to the checkbox labeled, "Internet Protocol Version 4 (TCP/IPv4) and click the "Properties" button.
C. Click "Advanced..."
D. Under "Default gateways", select the row "25.0.0.0" (or it may say "25.25.25.25") and click "Remove".
E. Click "OK" and close the Properties window.
7. From the adapters screen from step 6, ensure that there are no other third-party VPN adapters running. For example, "Wippien" is a similar (but inferior) application to Hamachi. If you have such an adapter enabled, right-click it and select "Disable". Unless these are disabled it will cause conflicts with Hamachi.
8. In your Windows 10 Search box, search for "cmd" and hit enter. In the command line, type: "ipconfig /flushdns" and hit enter.

You should now be able to right-click on another player's computer in your Hamachi network and click "ping". If your command window shows a list of "Reply" messages, you should be good to go. If not, try restarting your Hamachi network adapter (Disable, then Enable), and restarting the Hamachi Service in troubleshooting below.

If everything has been followed above for ALL PLAYERS, you should be able to play without a hiccup.

Troubleshooting.

Q: I can't get Hamachi to start, clicking the icon doesn't work!
A: In your Windows 10 Search box, search for "services" and open the application. When the Services window comes up, scroll down to "LogMeIn Hamachi Tunneling Engine" and click on it. On the left pane, click either "Start" or "Restart". Hamachi should come up fine now.

If you're still having problems, try these two official guides below. They helped.
https://help.logmein.com/articles/en_US/FAQ/Why-do-I-get-Request-Timed-Out-when-trying-to-ping-a-computer-en1
https://help.logmein.com/articles/en_US/FAQ/Resolving-Hamachi-Request-Timed-Out

I didn't touch on Port Forwarding, but that shouldn't affect you with Hamachi. Blizzard lists the required ports, but I'll leave that for someone else to guide through how to do. One last tip: Avoid running your LAN session over a mobile hotspot. Routing works very different and you will have problems.

Linux

The following was setup using Linux Mint. Following these steps I was able to play games with others, even those on Windows.

Most modern Linux distros include a Software Manager/Repository where you can use the GUI to easily download and install, update, and remove software. If you're a command line noob like me, I wholeheartedly recommend this.

From your Software Manager:
1. Download and install Hamachi
2. Download and install Haguichi (a Hamachi front-end GUI)
3. Download and install UFW (The Uncomplicated Firewall)
4. Download and install GUFW (A UFW front-end GUI)
5. Run Haguichi and setup your network just as you would in Windows.
6. When you're ready to play, run GUFW and turn Status to "OFF".
7. Remember to turn Status to "ON" again when you're finished playing.

That's It! You should be good to go.

Thanks for the guide. Have you had any luck running hamachi on the hoste OS and then the game within a virtual machine?

I've never tried it, but assuming your VM has full internet access, I would think Hamachi would function no differently. That's just my speculation though

I have never heard of such a Hamachi service before, thanks for the detailed manual. I think I'll try it. I tried many different virtual private network services but many of them did not work stably. Now I use service VeePN on all my devices and I am pleased with the quality of the service. But always need to have a backup option, you may also need it.

That's not true. Wippien is actually a lot better than Hamachi. Hamachi is old, bloated and bad implemented. No wonder why memory leaks and relayed tunnel are common issues on it.
Wippien is well designed and implemented. It only relies on direct connections (unlike Hamachi), no members limit, works with Jabber/XMPP accounts, highly customizable and configurable, free and open-source, very stable, reliable and lightweight. Plus, most things on it works remotely, not locally. So, you should not worry that much for it being an outdated software. As it mostly works remotely, if you ever need to reinstall your OS or gotta use it on another machine, you won't have to re-add your contacts and groups (networks). In short, take Wippien as a MSN Messenger with P2P VPN support.

With the above being said, if you want a decent cross-platform P2P VPN software, look for ZeroTier One. Wippien is also available for GNU/Linux. But, ZeroTier One is available for a much wider range of operating systems.

To each his own, however, in my experience Wippien has been far more unreliable than Hamachi. For one thing, Wippien uses a Jabber ID which requires you to link your personal email account to it in order to run a network. In order to do that, you will need to give your personal email address public access (disabling your security protocols) to your account for third party software to access it. That's a big red flag for me, and it means you'll need to remember to turn this on and off each time you want to play a game over VPN.

Furthermore, I've noticed that Wippien's app "friend list" is buggy. It lists people from your email account that you would never want to game with and I've had a hard time trying to clean them out. I wouldn't want to accidently send an IM to my boss about Diablo.

Thanks for the guide.

You don't need to (and shouldn't) provide your real email (Gmail) address to use Wippien. Simply create a Jabber account and use on it.
Here is, a decent XMPP service. That's plausible, if you've used your Gmail account. I've never had such issues with it.
At the time of Wippien's latest release, Google Talk was Google's main chatting service. As it was a standardized XMPP chatting service, Wippien used to work fine with it. It may have issues nowadays, because Google Hangouts (Google Talk's replacement) is not a standardized XMPP service.

I've got my PC and my craptop this far, even got the pinging to successfully reach each device. I can even see the game in the UDP listing. However, I still cannot successfully get the game to join. Any idea what went wrong?

EDIT: You know what I second the Radmin VPN. I just got the game working successfully through that :)

If you want to play on local network, you shouldn't use a P2P VPN software, Phoenix7786.

In fact, what you need to set up a network match is IPXwrapper. Because DirectPlay is an old API, it's not recommended to rely on TCP/IP connections.
IPXwrapper simulates an IPX hardware, so that you can play old games. Although not all DirectPlay-based games have support for serial connections, Virtual Serial Ports Emulator is another option, which simulates serial hardware in many different ways, and is a more convenient software for those who prefer port-forwarding over P2P VPNs.

However, if you indeed want to play outside your network, consider using n2n or Wippien. Both n2n and Wippien are free and open-source P2P VPN software (Wippien is not GPL'd, copyleft software, though).

EDIT: Sorry for the messy text, the BBcode from GOG is terrible.

Virtual Serial Ports Emulator can be downloaded from here: https://web.archive.org/web/20110710203301/eterlogic.com/downloads/SetupVSPE.zip
IPXwrapper can be downloaded from here: https://www.solemnwarning.net/ipxwrapper
n2n can be downloaded from here: https://meshvpn.blogspot.com/2020/12/n2n-ntop.html
Wippien can be downloaded from here: https://meshvpn.blogspot.com/2019/04/wippien-flawless-alternative-to-hamachi.html

Sorry for the late revival of this post. But I thought it might be worth mentioning here that we have added ZeroTier integration in to DevilutionX, which works similarly to Hamachi but since it is build in there is ZERO configuration, you simply create your game and it will show on the game list of the other player (or you can create one with a private game with a password, but still no need for any router or software configurations).

Happy Diablo-ing every one :)