Where can you find Legendary gear, and is there anything you can do to increase the drop/drop rate?

The simple answer is: it's totally random.

But it's actually not quite just that simple. There is a built-in "bad luck protection" that slightly (it is unknown by how much) increases your chance to get a legendary each time you don't. Some examples of sources of legendaries include:

• Daily world quest caches
• Bosses in Mythic dungeons
• Weekly Mythic+ cache
• Raid bosses
• Withered training end chest
• PvP strongboxes

You can look up information on specific legendaries at WoWdb. From there you can find the drop rates of specific legendaries and what drops them. For example Kazzak's Final Curse has an estimated 0.003% drop rate from the Wrath of Azshara in Eye of Azshara, 0.3% drop rate from looting the Peerless Challenger's Cache in Halls of Valor and 0.007% drop rate from Kirin Tor Chest in from the Kirin Tor of Dalaran world quests.

As for the drop rates, you have experienced how rare getting a legendary may be. Although Nathan mentions the "bad luck protection", it is still random to get a legendary drop. Each time you complete a task for a legandary, such as killing an NPC or completing a dungeon, there is a dice roll for items in the item drop pool. Each time you roll is independent of the other, meaning an item with a 0.2% drop chance will have a 0.2% chance to be looted each time you loot the item. (The "bad luck protection" is noticeable in the long term but short term is not.)