To purchase Diamonds, first tap "Store" on the main menu.
Your Diamond total is displayed at the top of the screen. Tap the green "+" icon next to it.
This will take you to a screen which offers numerous different packages.
Tap the price button under the package you would like to purchase, then follow the instructions to make the purchase and acquire the Diamonds.
If you have bought Diamonds, but did not receive them, read this article.