The two covenants that are the fastest to reach maximum rank are either Heirs of the Sun or Way of Blue.
Heirs of the Sun
If you join Heirs of the Sun, you are awarded a Sunlight Medal anytime you have successful (jolly!) co-operation. You can achieve that through either of the White Sign Soapstones. Using the standard one will require you to defeat the boss without either you or the host dying; however if you use the small stone all you have to do is keep you and the host alive until you reach the time / kill limit.
An alternate way of getting Sunlight Medals is to summon people who are already in Heirs of the Sun; their summon signs are a much brighter orange and their characters are orange / gold as opposed to the standard white. For each covenant member you summon and are successful with, you will be awarded. This route is helpful early on as accessing this covenant isn't until a significant way through the game, and you do not need to be a member to be awarded these medals. You will need 30 medals in order to reach Rank 3.
Way of Blue
This is the first covenant you can join in the game. You rank up in this covenant by defeating invaders (I'm not sure if NPC invaders count towards rank, but I'm guessing they don't). You will need to defeat 10 invaders to reach Rank 3.
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From my personal experience, I chose the Heirs of the Sun route as it is very easy to be summoned at every boss relatively frequently (especially as you progress further into the game). Put your sign down 2-3 times at each boss and you should have 30 medals in no time. These are the bosses I had the easiest time to be summoned for:
*** Spoiler - click to reveal ***- Ruin Sentinels (Lost Bastille)
- The Lost Sinner (Sinner's Rise)
- Executioner's Chariot (Huntsman Copse)
- Smelter Demon (Iron Keep)
- The Duke's Dear Freja (Brightstone Cove Tseldora)
- Looking Glass Knight (Drangleic Castle)
Even just sticking to these bosses, you'll have no problem getting enough medals.