In the fourth generation, a variety of things have changed in order to allow for the obtaining of Shiny Pokémon. First is a method commonly known as the Masuda Method, named after the game developer and the person who revealed it; Junichi Masuda. This method has you breed two Pokémon. However, one of the Pokémon must be of a different nationality than your game (such as a Japanese Pokémon on an English game). This will lower the chances of hatching a shiny Pokémon from 1 in 8,192 to 1 in 2,048 cutting it by 75% in Generation IV. However, in Generation V, it is boosted even further from 1 in 8,192 to 1 in 1,365.3. Generation VI has increased the likelihood even further, with the rate being reduced even further.
The second method of obtaining shiny Pokémon is through the method called "chaining". This method has you carry on a chain on the PokéRadar within Diamond, Pearl, Platinum, X & Y. As you battle the Pokémon and your chain increases, the chances of seeing a Shiny Pokémon also increases up until you're on the 40th chain where it levels out. Continue the chain and you may see the grass glow instead of shaking. If this occurs, there is a shiny Pokémon there. When you defeat or capture a Pokémon you have discovered with the PokéRadar, the chain begins and you need to follow the chain to other grass tiles to continue the chain. These chains are broken when a different Pokémon appears or you encounter a Wild Pokémon outside of the set grass tiles so it is recommended that you use Repel or other Wild Pokémon prevention techniques. Shiny Pokémon are easily findable with the PokéRadar. If you have found a Shiny Pokémon with the PokéRadar, the grass will shine as opposed to shaking. With chaining, you have the ability to increase the likelihood of finding a Shiny Pokémon. This is done by having a prolonged chain on the PokéRadar. The chances increase up until the 40th Pokémon in the chain. This is calculated with the following formula;
Chance = (14747 - 40*ChainNumber)/2621440(41-ChainNumber)
After each use of the PokéRadar, it requires recharging. This is done by walking 50 steps.
Chaining goes even further in Pokémon X & Y due to the inclusion of Consecutive Fishing. With this, as you hook Pokémon with the fishing rod, you get increased chances of getting a shiny Pokémon. This will increase with each Pokémon you hook without leaving the spot, regardless of if you defeat them, capture them or run from battle. If you don't get a nibble, or if you move from the place you were fishing, you will break the chain. The chain also resets once you get a shiny Pokémon. Exact values are still to be discovered.
In Pokémon Black 2 & White 2, a new item is introduced called the Shiny Charm. This item, obtained when you have completed the National Pokédex, increases the chances of Shiny Pokémon appearing. As it's a Key Item, when you have the item in your bag, then the chances of encountering Shiny Pokémon in the wild is increased to 300% of normal, making it a 1 in 2,730.6 chance. If, however, you're using the Masuda Method to breed, it reduces it from 1 in 1,365.3 down to 1 in 1024.
The best way would be using a PokéRadar