What are the requirements to get supply drops?

Solution 1:

Supply drop's rarity

55% chance of item being enlisted
30% chance of item being professional
15% chance of item being elite

Supply drop's item's

35% will be weapons
20% will be reinforcements
45% will be apparel

The only way to increase the rate in which Supply Drops are given is to play Multiplayer matches. The more often a player participates in Multiplayer matches, the more often Supply Drops drop.

A way to test this is to count how my games you play each day over a week, compare that with how many supply drops you receive within that week, then do the same again for another week and then compare the two results. The test would need to be fair, exact same amount of games played each week and completing all games.

So if this is true technically the amount of supply drops you receive is solely based upon the amount of games you play in multiplayer and NOT how you perform.

You can earn more item's like you'd receive in a supply drop by completing specific objectives within the Advanced Warfare single player campaign.
(Within the campaign menu, you should be able to see these objectives)

You can also receive items like you would in a supply drop by playing Exo-survival, there will be similar objectives in Exo-survival like the campaign.

Sources:
Activision
Advanced Warfare
Reddit