Haven't been on the Breakaway yet but based on other NCL ships:
No water drinking fountains (at least I don't recall ever seeing one on any ship).
Tap water in your stateroom is drinkable (reverse osmosis filtration).
You can ask for a glass of water at any restaurant or bar.
Bottled water is available for purchase or you can bring your own as carry-on or checked luggage.
Top 10 Contributor
The water onboard the ship is safe to drink. The ship has a water purification system that ensures that the water is safe to drink. The water is used in the restaurants, kitchen, bars and other places on the ship.
However, if you prefer, you can always bring cases of bottled water onboard with you. You can check the water cases along with your luggage and it will be delivered to your room.
Number One Contributor
The rooms come with water bottles (but you have to pay for them)
I've drank the water from the faucet but would do so as a last resort.
I usually travel with a stainless steel bottle. (no chances of stateroom attendants throwing away your bottle by accident because it's plastic - happen to me once). At the drink stations where the buffets are there is cold water available. They will usually have signs saying not to fill your water bottle there. But you can take one of the cups and use that to fill your travel bottle that way. Best way to stay hydrated during a cruise.
Happy trails :)