The Warlock Zoo deck is commonly considered to be the cheapest competitive deck. It costs around 1200 dust to craft, the latest version from Reynad which I'm currently trying out has the following decklist:
Most of the minions used in this deck are pretty good on their own, crafting them won't lose you that much even if you don't continue to use this specific deck later. Doomguard is rather specific to this deck, but I wouldn't try to play this deck without it.
Here's a video of Reynad explaining the idea behind the deck, there are also some example games in the same channel:
If you just started playing and don't want to or can't craft the entire deck, I'd recommend to start with a variant of Trump's Free Mage. The Mage class has very good class cards like Fireball and Water Elemental, and Polymorph can help a lot against the big legendaries which you'll inevitably encounter at some point.
You can leave out the more expensive stuff and replace it, the deck will be a bit weaker but it should still work. I'd craft the Harvest Golems and the Azure Drake first, they're really useful in many decks. I also like the Amani Berserkers, and they're especially good with a Mage that can easily enrage them with the Hero power.