What is the best way to improve my collection without spending money?
The ABSOLUTE best way to improve your collection is the following:
Do your daily quests! Tim S. already mentioned this, but you should try to reset your quests that give below 60 gold in hopes of a better one.
Another way to obtain gold is by winning 3 matches. Every 3 matches you win earns you 10 gold up to 100 gold per day.
Use every 150 gold you earn to play arena! This is where my advice differs from Tim. If you win at least 3 victories in arena you will generally net at least as much as your investment of 150 gold. If you win more than 3 victories you get even more than your investment! You can see a list of all possible rewards for all possible number of victories here: http://hearthstone.gamepedia.com/Arena
If you are concerned about your ability to pick cards during the arena draft there are numerous resources available on the internet that will help guide you through the process. Just google "hearthstone arena tier list".
Arena can also be a great learning experience. There are lots of decks out there that are very competitive that are also fairly cheap dust wise. Learning card relationships and value through arena can help you not only learn how to play your decks but help you learn to develop your own! Hope this helps!
// addition
I've seen a few people disagreeing with the number of wins that I put as the break even point for arena. So I would like to defend my position on that front.
If you look at the arena rewards list you can see that it's possible to break even at 2 wins (card pack + potential reward of 40-60 gold). It's even possible, in terms of gold, not to break even at 5 wins (card pack + reward of 45-60 gold + common card). I am uncertain as to what the odds are for each item in the random reward pool, but I have noticed that gold is the most common.
So the reason I list 3 wins as the break even point is because after experiencing many 3 win arena runs I would expect my average gold return to be in the 140-145 range. Once you include the average dust earned, lets just say I have never felt bad about a 3 win arena run.
I guess my point is, on average, a 3 win arena run will net you close to the amount of gold you invested (if not the total original investment). Plus once you include the average dust earned per 3 win run, the average value you received is easily worth the 150 gold you invested.
Like @ThomasP said, but I want to add the following:
If you're going to play ranked in the long run, craft early on one good cheap deck (usually these are aggro decks, like Pirate Warrior) and try to reach rank 5 every month. That alone gives you 505 dust (1 golden epic, 2 golden commons, 5 dust) for more crafting. Preferably, this deck should have several cards in common with other archetypes. I say one cheap deck because crafting cards is usually a waste of dust if you later open the same card from a booster, so you should only craft cards that you're going to use right away.
You should balance Arena runs with buying Adventures (700 gold each wing). Adventures give generally good cards that you can't get anywhere else. With the coming Standard change you won't be able to buy Naxxramas with gold unless you have the first wing, so I would buy that one at least. If you don't want Naxxramas cards you can disenchant them when Standard arrives.
Also, when Standard arrives there are going to be some balance changes which could affect Classic cards, so I wouldn't disenchant many cards until they announce the changes. Why? Because when they make an errata you can disenchant for the full crafting cost (50 dust for a common instead of 5).
And always get the free booster from weekly Tavern Brawl.
Edit: replaced old Face Hunter list with a post Gadgetzan Pirate Warrior
Personally I think the best way is to complete the daily quests as often as you can, replacing the lower gold ones (clicking on the "x" to get a new one) for hopefully more valuable ones. Every 100 gold, buy a pack. Also make sure that you win at least one Tavern Brawl each time it's available, as you get a free pack for that as well.
I used to save up the gold for for the dungeons, but I did some research and found out that the cards I would receive in most cases weren't worth the cost. (Except for the Grim Patrons so far), but I've not yet researched all of them.
Every time you open a pack, you of course have the chance of getting a card you already have 2 of, so the 3rd one is useless. This should be turned into dust. After this happens often enough, you should be able to craft some pretty decent cards.
Regarding legendaries vs rares, it depends on what kinds of cards you need. If the legendary is going to be super useful, craft that, otherwise craft something you think will round out your deck. Some cards have much more utility than others, but it also depends on what kind of deck you're building.
I also tried the arena, but found out that because I didn't have a stellar deck, I would lose a lot and it wasn't worth the gold investment. Personally I wouldn't try the arena again until I could get at least to rank 10 or higher, symbolizing that not only did I have a pretty good deck, but I knew how to play it effectively.
Apparently my experience with the arena (as well as my memory of it) wasn't that great, so if you're pretty good at playing a particular class and win consistently, it might be worth a go. For me, the other option of completing quests and using the gold to buy more packs was more certain though.
TL;DR: I'd buy packs, unless you have a phenomenal deck already, then I'd try arenas.