Dragon-type Pokémon are really powerful. Aside from the fact there's lots of Legendary and Pseudo-Legendary ones, there's several reasons to like them.
Dragon-type moves work well against almost any opponent. Only Steel-types (and now Fairy-types) are a real problem to take down.
They also have few weaknesses themselves, only Ice-types, Fairy-types, and other Dragon-types.
Now, the conventional wisdom is that you need at least a somewhat balanced Team, or else your opponent will be able to exploit your Pokémon's common weaknesses.
However, just like Gym Leaders, Elite Four, and some Champions (Lance, Iris, and Wallace), it IS possible to focus on a single type, and Dragon-types are (IMO) the best type to do so with.

Now, here's some advice on how to do this:

1. Have a well-balanced Team
That's right, have a well-balanced Team of just Dragon-types.
Just because all of your Pokémon are Dragons doesn't mean they all have to have the same secondary types.
You can even add some non-Dragon-type Dragons to your Team (Charizard, Gyarados, Tyranitar, etc.), if you want to.
Both Lance and Iris do this (Lance doesn't in the PWT, though), and it can help in dealing with Ice-types immensely.

2. Diversify your moves
Here's an example:
A Salamence (holding a Salamencite) that knows Draco Meteor, Thrash, Fire Blast, and Dragon Dance.
If facing a Steel-type you can use Fire Blast.
If facing a Fairy-type, Thrash should work well (Remember: after Mega Evolving, Thrash is a Flying-type move, and does 50% more damage).
And although Draco Meteor quickly decreases in strength, you can just switch to using Thrash afterwards (that technique also works with Outrage, Dragon Ascent, or any other physical move).
You'll always want make sure you have some powerful move(s) to use against Steel-types and Fairy-types (and something that works on Shedinjas, too).

3. Using the right Held Items (and having the right Abilities, too.)
A Dragonite with the Ability Multiscale (If you're lucky enough to have one), holding onto a Yache Berry, will take ¼ damage from an Ice-type move, assuming its hit when its at full HP. Now that's powerful!
Yache, Roseli, and Haban Berries are extremely useful items, as they can cover your weaknesses.
Alternatively, you could use the very powerful Weakness Policy (which I regularly have those same Multiscale Dragonites hold onto), for a more aggressive strategy.
The White Herb works well with Draco Meteor, and so does Dragonium Z (If you turn Draco Meteor into a Z-move, it doesn't lower your Sp. Attack).
The best Z-Crystals for a Dragon Master, however, are Kommonium Z (let's Kommo-o use its Z-move "Clangorous Soulblaze", which raises all of its stats), and Ultra Necrozium Z, which lets Dawn Wings or Dusk Mane Necrozma use Ultra Burst, which is similar to (but doesn't conflict with) Mega Evolution.
Then, Ultra Necrozma (now a Dragon-type) can use its Z-move "The Light that burns the Sky", which will either be Physical or Special, depending upon its current stats.
It should be noted here that Rayquaza can Mega Evolve without holding a Mega Stone, which means Mega Rayquaza can use any Held Item (other than Z-Crystals).
In Double and Triple Battles, Mega Rayquaza's Delta Stream Ability can be extremely useful for protecting Flying-type Dragons, like Dragonite, Salamence, and itself, against both Ice-type and Rock-type moves (as well as Electric-type moves, for what its worth).

4. Focus on Power
A Dragon Master should ALWAYS be trying to take their opponent out in one hit.
Anything you can do to make your Dragons more like One Punch Man is good.
This means focusing on raising your Attack, Sp. Attack, and/or Speed, depending on the Dragon in question.
Additionally, you'll want to make use of moves like Dragon Dance, Swords Dance, and Shell Smash.
I actually defeated Cynthia with a Level 51 Gyarados, simply by maxing my Attack and Speed with Dragon Dance (had to heal a few times during that), and then taking each of her Pokémon out in one hit.
Now, when you can't take them out in 1 hit, it helps to have priority moves to finish them off. Dragonite, Rayquaza, Zygarde, and Arceus (which can be a Dragon-type) all can learn Extreme Speed, and Giratina can learn Shadow Sneak, so there's plenty of choices (there's probably even more than these).
Also, although moves like Dragon Claw and Dragon Pulse are reliable, consider more aggressive, higher Power moves.
I mean, Outrage is a 120 Power, 100 Accuracy move, regardless of its downsides.
The same goes for the 130 Power, 90 Accuracy move, Draco Meteor.
And like I mentioned above, these two moves, learnable by pretty much all Dragon-types, make a great combo (Clair's Dragonite and Salamence use it in the PWT).

Anyway, that's all the advice I have for now, other than to study the Teams and Movesets of other Dragon Masters, such as Lance, Clair, Zinnia, Iris, and Drake (just to name a few).
I hope this helps those who want to be a Dragon Master.