Start here on NicoNico. Note there is often a discount for ordering shows early. Korakuen Hall shows are traditionally 1500 points/yen and the big Pay Per Views are at 2160 points/yen for a preorder and 2700 pts/yen day of show.
Overall, the process is very easy:
1- Sign up for a free account. You can Google Translate the page to make it easier, but the registration is fairly standard, so it’s not really needed.
2 - Here's the Dragon Gate specific page. 1 point = 1 yen. Upcoming shows are down at the bottom.
3 - Go here to buy points. Note that any that you have left over after your purchase can be used the next time.
4 - When you click on the points you want to buy, you’ll be presented with several payment options. As you proceed, it’s the gold buttons with Japanese writing that confirm each step (the grey button is cancel).
5 - Once you’ve bought your points, go back to the show page. Click the big red button near the bottom. That will confirm that you’ve bought the show with X of your Y points, and you have Z left:
6 - You have the ability now to watch live. If you want to “timeshift” it, simply hit the yellow button that says Timeshift which appears on the show page and where the show is listed in your account. This will allow you to watch on demand for a week or so after the event.
7 - To be sure you’ve done everything right, go here. It should look like this:
That's all there is to it. It took me about 3 minutes for the whole process and it was very easy. The best news for me is that they have EVERY Dragon Gate show both live and on demand, including all the Korakuens, alll the TVs and all the big shows!