There are two locations to brew ales at: upstairs of the Laughing Miner Pub in Eastern Keldagrim and the basement of The Green Ghost inn in Port Phasmatys. Start with the vat in Keldagrim.
Fill your inventory with 2 empty buckets or buckets of water, 2 barley malts, 1 package of "The stuff", 4 clean harralander, two empty pots or pots of ale yeast (there is a pot respawn in Keldagrim), and 2 Calquat kegs. Head to the vat in Eastern Keldagrim via the trapdoor northwest of the Grand Exchange for the quickest access. The Laughing Miner Pub is located north of the mine carts, and northwest of the pub is a well to fill your empty buckets with water if they aren't already filled. Head to the pub and go upstairs to the brewery. Pick up two pots from the pot respawn near the vat if you don't have them already. Talk to Blandebir to fill up your two empty pots with ale yeast for 25 coins each. After you have all the necessary items, fill up the vat by putting in the two buckets of water, two barley malts, The Stuff, four clean harralanders and one pot of ale yeast. You must fill the vat in the correct order - don't forget The Stuff, it increases the chance of mature ale, which is worth more, to 50%. After you put the ale yeast in, the ale will begin to ferment. The vat in Keldagrim can take 8 hours to 2 days to finish fermenting.
Head back to the Grand Exchange via the mine carts, and withdraw the same items once again to fill the vat in Port Phasmatys. Withdraw an Ectophial for the quickest access to the brewery, 2 barley malts, 1 package of "The stuff", 4 clean harralander, and 2 Calquat kegs. You should have 2 empty buckets and 1 pot of ale yeast in your inventory from your earlier trip to Keldagrim. Use your Ectophial and head to The Green Ghost inn, southeast of the Ectofuntus. Go downstairs to the brewery, and use the tap on the far wall to fill up your empty buckets with water. Repeat the vat-filling process by putting in the two buckets of water, two barley malts, The Stuff, four clean harralanders and the last pot of ale yeast, in that order. After you put the ale yeast in, the ale will begin to ferment. The vat in Port Phasmatys takes a minimum of 2 days to ferment, but rarely exceeds that.
When a vat of greenman's ale is done, the examine text will say "This vat is filled with [mature] Greenman's ale." Click the valve on the vat to empty the vat into the nearby barrel, then use your Calquat keg on the barrel to collect the ale. If you check the Port Phasmatys ale two days after it begins fermenting to see if it's done, you can go through the same vat-filling process to collect and refill the ale vats, starting with Keldagrim, in one go.