Hi I've only recently started playing Magic and have gotten a good feel of the game. I've made about 3 decks of my own and they've been pretty good but nothing excellent. I'm hoping that my new burn deck will be one of those "excellent" decks.
 

With my new burn deck, what I'm focusing on is killing the opponent will full-on burn and the odd creature management.

I'm hoping to have burnt them to the ground within turn 6 or 7. All of my burn cards cost very little. The main problem I think I have is that when I've depleted my hand, I can only pressure the opponent with one card at a time.
 

At the moment, I've only got 42 cards. I was hoping that you could tell  me what cards to keep and what cards I should take out and what additions I should make to my 42 card deck.
 

Anyway, here's my decklist.

Lands- 14

14x Mountain

Instants/Sorceries- 28

4x Shock

4x Dead//Gone

4x Lightning Axe

4x Firebolt

4x Lava Spike

4x Volcanic Hammer

4x Pulse of the Forge

P.S. Could you please keep the cost of the cards low as I have a small budget to work with.
 

It doesn't get any more straightforward than this, does it? Tap mana, play a card, deal damage to opponent. Well, it used to be a straightforward idea, anyway. I remember when I first started playing Magic back in 4th edition, if you wanted to build a burn deck, there was Lightning Bolt, Fireball, and the pseudo-burn Ball Lightning. Okay, there were a few others, but the point is that the choices were few, making the options easy. Today, however, that is a different story.

There are several different directions you could take this, even just keeping with semi-modern cards (and by the way, when people don't mention what format of deck they are building, I usually keep it to Extended-legal cards - currently back through Invasion & 7th edition). But, keeping this deck as close to it's current theme as I can, and keeping to inexpensive cards, I do have a few ideas, starting with the basic:

Seal of Fire: Deal 2 damage for 1 mana. It's almost like another Shock at a slower speed.

Flame Burst: The first time you play this, it's only 2 damage for 2 mana - not especially great. But the next time you play one, you get a Volcanic Hammer, and it only gets better from there.

Rift Bolt: There's no reason (usually) not to suspend this for one turn and deal 3 damage for one mana.

Flames of the Blood Hand: Dealing 4 damage while only spending 1 card and 3 mana is a good deal. This can also mess with those annoying players who like to prevent damage and gain life. A good addition.

Now, you mentioned having the problem of running out of cards in your hand. That's a common problem of red burn decks, and it's called topdecking. You've already got 2 cards that can be (potentially) reused in Pulse of the Forge and Firebolt. These are both excellent, and about the best we can probably do with reusable damage without getting into some expensive rares (though I will mention a few choice rares at the end should you happen to come across some or decide to splurge a little). So, the next best thing is to draw more cards. My choice here is a rare, but you should be able to get a set of 4 for well under $10: Wheel of Fate. Ideally, you can suspend the Wheel on turn 2, and 4 turns later - about the time your hand is running out, you can get a hand full of more burn to finish the opponent off!

Another option is to use a couple "X" spells, such as Fireball or Disintegrate. These won't be much help on the first few turns, but might be that perfect way to finish someone off once you have several Mountains in play.

For a little different feel, you could go the direction of Grapeshot. The idea here would be to hold onto your burn spells and just play your land. Then, when you get to, say, 5 mana, drop something like 2 Shocks and a Lava Spike, then play a Grapeshot for 4 damage, making for a pretty big turn. The potential problem here is that if you can't get a storm count going, Grapeshot is a pretty sub-par card. 

Now, for what you had in your deck originally. Your deck is looking to throw 20 damage at the opponent as quickly as possible, so you need all your damage cards to be able to target the opponent. Sometimes you'll want to take out a creature that threatens to kill you before you can kill your opponent, but the cards that only target creatures should probably go, so I would suggest dropping Lightning Axe and Dead//Gone.

As far as land, I'm going to go out on a limb and break one of my own rules here and say this deck should run well with 18-20 lands. Most of the time that's not enough, but with only one color and no spells that cost more than three (other than potentially a Fireball), it should work out. Just the same, if you have problems getting that three mana quickly more than once in a rare while, don't be afraid to add a couple more Mountains.

So, if I were going to play this deck, I would probably start with a decklist something like this...

Lands- 20

20x Mountain

Instants/Sorceries- 40

4x Shock

4x Firebolt

4x Lava Spike

4x Volcanic Hammer

4x Pulse of the Forge

4x Seal of Fire

4x Flame Burst

4x Rift Bolt

4x Flames of the Blood Hand

2x Fireball

2x Wheel of Fate

...and I would play at least 20 games with it. After that amount of time, I should have a pretty good idea what's working and what isn't. Only after that point would I work on adjusting it more.

Now, for those exciting rares I mentioned earlier. If you get a lucky pull from a booster or a good deal on a trade, here are some cards you could look out for:

Magus of the Scroll: This is great reusable damage in the form of a creature. Your hand is often going to be low on cards, so picking the named card shouldn't be too hard.

Char: Another 4 damage for 3 mana spell. Don't worry about it doing 2 damage to you - that won't often matter.

Hammer of Bogardan: Another reusable damage source. Anytime you have extra mana, just bring this back in your upkeep and use it again.

Jaya Ballard, Task Mage: Did you just draw another useless Mountain? Discard it and turn it into 3 damage!

Browbeat: Your opponent has a tough choice: Take 5 damage now, or let you draw (potentially) 3 more burn spells. Hmmm....

Fiery Gambit: If you're a gambler, this is your card. 50% of the time it does nothing, but it has the potential of dealing 3 or 6 damage, or even untapping all your lands and letting you draw 9 (yes, 9) cards!

I hope this gives you something to go on! Happy burning!