Is it possible to get a duplicate legendary from the Frozen Throne prologue

According to what most people that opened a lot of packs said, there's quite a low chance of getting a legendary.

On average, people get 1 legendary per 20-25 packs. Some are less fortunate than that however.

Source: S1, S2, S3

An answer to your question depends on what you are trying to get out of the game.

Collecting all of the Cards!

According to the Economy of Hearthstone, in order to collect all of the cards, you should save all of your dust until you have enough to craft the cards you have missing. This article simulates optimal dust usage to calculate the total number of packs you would need on average to collect all of the cards.

Creating the Best Deck

Creating an excellent deck does not require a lot of dust or legendary cards. Trump was able to get to the Legendary rank very quickly without spending any money--you can see it here. In this series he spends his dust on several commons, a few rares, and a single epic (if I remember correctly). Most of the top players have a few legendaries in their decks, but it is not required. Decide on a deck you want to create and spend your dust getting those cards. The optimal way to spend dust to create a particular deck as fast as you can is to save up your dust until you can craft the rest of the deck.

Creating a Decent Ranked Deck

Perhaps you just want to play ranked and "rank-up" relatively quickly. The best way to do this is not to attempt to build the best deck which lots of rare cards, but to build a good deck with strong cards.

1. First, decide on a deck strategy that relies mostly on rares and commons. 90% or so of your deck will be made of these, so they have to work on their own with support from epics and legendaries.
2. Second, fill out the deck with cards that you already have that would work well in your dream deck.
3. Third, fill the rest of the deck with cards you already have that may not be your ideal picks, but are at least good.
4. Fourth, play with your not ideal deck. For optimal dust usage, you want to determine what is missing from your deck and this is best done through experience rather than theory-crafting.
5. Spend your dust on the cards your play experience dictates would be the best additions. If you can't afford all of your ideal cards, spend your dust getting the largest number of ideal cards you can. This is an optimal use of your dust because it gets your deck as close as possible to your ideal deck.