Aggro and zoo decks (hunter warlock shaman) are the most effective budget decks. Sure they can benefit from a few select rares/epics and adventure cards, but they work. The unfortunate thing is they can be countered, and sometimes you will just run into continuous anti-aggro and feel like you can't win anything. It sucks, but it happens. Just take a break for the day and try again later.
The silver lining is that your matches should be short, and if you aggressively reroll for quests tailored for your archetype you can make steady gold progress each day.
If you do decide to spend actual money, adventures are more likely to give you playable high value cards than random packs. League is generally recommended, since Naxx is being phased out. BRM is also fine, but only truly neccessary for very specific decks.