There are many ways to get The Ur-Dragon onto the battlefield no later than your 4th turn. I'm only going to detail a few of them to give some ideas of what to do.
First, you have a snowball's chance in hell of making a 1st-turn The Ur-Dragon happen, but just in case, here's how to do it:
[ATTACKING WITH THE UR-DRAGON ON YOUR 1ST TURN]
Opening Hand: Cascading Cataracts, Channel, Elvish Spirit Guide, any equipment that gives haste to a creature, Lotus Petal, Utvara Hellkite (optional), Sol Ring (optional).
1) Play Cascading Cataracts and Lotus Petal.
2) Tap and sac Lotus Petal for 1 green mana. Exile Elvish Spirit Guide from your hand for 1 green mana.
3) Use the 2 green mana to play Channel.
3a) If you have Sol Ring then pay an amount of life equal to 8 plus the mana cost and equip cost of the equipment you have that gives a creature haste.
3b) If you don't have Sol Ring then pay an amount of life equal to 9 plus the mana cost and equip cost of the equipment you have that gives a creature haste.
Optional: If you have Sol Ring then use 1 colorless mana to play it. Tap it to get 2 colorless mana.
4) Filter 5 colorless mana through Cascading Cataracts to get one mana of each color. Use the mana of each color plus 4 colorless to play The Ur-Dragon.
5) Use the rest of the colorless to play your haste-giving equipment, equip it to The Ur-Dragon, and swing.
6) Including the draw from The Ur-Dragon's ability, you should have three cards in your hand. If one is a permanent then play it with his ability. If it happens to be Utvara Hellkite then you have a 3rd-turn win on your hands: Turn one, attack with Ur-Dragon and play Utvara Hellkite, turn two, attack with Ur-Dragon and Utvara Hellkite, get two 6/6 dragon tokens and a dragon card into play from your hand, turn three, attack with Ur-Dragon, Utvara Hellkite, two 6/6 dragon tokens, and the dragon you played with Ur-Dragon's ability on the previous turn, dealing 40+ damage total from all attacks over the three turns.
[ATTACKING WITH THE UR-DRAGON ON YOUR 2ND TURN]
Method 1) Same as the 1st turn method, but you can have a land that produces a green mana and either the Lotus Petal or Elvish Spirit Guide, but both aren't necessary.
Method 2)
2.1) Play a land, tap it for mana, exile Elvish Spirit Guide from your hand for a green mana, play and sacrifice Lotus Petal for another green mana.
2.2) Use both green mana to play Channel and pay an amount of life equal to 5 plus the mana cost and equip cost of haste-giving equipment that you have to get the same amount of colorless mana.
2.3) Use the colorless mana to play Geode Golem and the haste-giving equipment, equip the equipment to Geode Golem, and attack with Geode Golem.
2.2) Play The Ur-Dragon when Geode Golem's ability triggers.
2.3) Attack with The Ur-Dragon during your next turn.
[ATTACKING WITH THE UR-DRAGON ON YOUR 3RD TURN]
Method 1)
1.1) On your 1st turn, play a land, use its mana to cast Sol Ring, and use the Sol Ring's mana to cast Star Compass.
1.2) On your 2nd turn, play a land, use the mana from your two lands, Sol Ring, and Star Compass to cast Gilded Lotus, use the mana from Gilded Lotus to play Mind Stone and Fellwar Stone, use the mana from Mind Stone and Fellwar Stone to play Expedition Map.
1.3) On your 3rd turn, use Expedition Map to get Tolarian Academy out of your library and into your hand, play Tolarian Academy, and tap it for 5 blue mana.
1.4) Between your two other lands, Star Compass, Fellwar Stone, and Gilded Lotus, you should be able to get the other four colors of mana you need, plus two extra mana from Gilded Lotus.
1.5) After all mana sources are tapped, your mana pool should contain something like BCCCUUUUUGRRRRW (15 mana total). Play the Ur-Dragon, haste-giving equipment, equip it to the Ur-Dragon and swing.
1.6) See Step 6 of attacking with the Ur-Dragon on the 1st turn.
Method 2)
2.1) On your 1st turn, play Lotus Petal, tap and sac it for 1 mana of any color (doesn't matter which color) to play Sol Ring, tap Sol Ring for 2 colorless mana to play Lightning Greaves or Swiftfoot Boots. Play Tolarian Academy as your land for the turn. Tap it for 2 blue mana to play Mind Stone, tap Mind Stone for 1 colorless mana to play Expedition Map.
2.2) On your second turn, tap Mind Stone for 1 colorless mana and play Elixir of Immortality. Tap Sol Ring for 2 colorless mana, activate Expedition Map, get an artifact land, and play it. Tap Tolarian Academy for 5 blue mana (Sol Ring, Lightning Greaves or Swiftfoot Boots, Mind Stone, Elixir of Immortality, and your artifact land make 5 blue mana), play Geode Golem, equip it with the Lightning Greaves or Swiftfoot Boots (using the mana from your artifact land) and swing.
2.3) Play The Ur-Dragon when Geode Golem's ability triggers.
2.4) Attack with the Ur-Dragon on your third turn.
There are variations and combinations of the different parts of these methods that you can use to get The Ur-Dragon onto the battlefield by your 4th turn. You just have to be creative and be able to improvise.
Beyond that, the idea is to get as many dragons on the battlefield as possible and go stompy. However, the deck does have a small bag of tricks just in case basic stompy isn't working out.
[BAG OF TRICKS]
Vault of the Archangel helps you regain life not just from taking damage, but also from paying life when using Channel.
You can use Scion of the Ur-Dragon to get dragons into your graveyard then use Bladewing the Risen to bring them into play.
With The Ur-Dragon putting cards in your hand, you should have plenty of Dragons to reveal for Kilnmouth when it enters the battlefield. In that case, you'll have some direct damage available if basic stompy is working out.
Scourge of Valkas and Dragon Tempest also provide some direct damage.
If you activate Vault of the Archangel then deal direct damage in the aforementioned ways then you can kill your opponents' creatures regardless of how tough they are and you gain life for doing it.
Direct damage can help clear creatures out of the way. Destructor Dragon and Scalelord Reckoner and help with removing other permanents. You can even deal direct damage to Destructor Dragon to kill it if you really need to get rid of something.
You should have enough creatures in play to defend Sarkhan long enough to build him up for his big pop. Once you get that to go off, Scourge of Valkas and/or Dragon Tempest and Lathliss, Dragon Queen, should make it possible to deal enough damage to kill all of your opponents due to so many dragons entering play at the same time.