To help get to 10 billion faster, you need to look to buy Stud Multipliers. There are 5 (2x, 4x, 6x, 8x & 10x) like all LEGO games. Applying them all together will create an even greater multiplier: 3840x.
•2x - Use the code 1F5YH2
•4x - in Amon Hen (first quest to the left of map statue); Craft Item needed is the Mithril Fishing Rod. The Red Brick costs 1,000,000.
•6x - located in Edoras (below the statue from map view); Craft Item needed is the Mithril Top Hat (located in The Battle of Pelennor Fields; after fight with The Witch King, blow up one of the Mithril LEGO Pieces); Red Brick costs 4,000,000.
•8x - located in Helm's Deep (to the left of the map stone from map view); Craft item needed is the Mithril Shield (located in The Path of the Dead; to left of final area before fight); Red Brick costs 8,000,000.
•10x - located in Cirith Ungol (to the left near two Mithril pieces); Craft Item needed is the Mithril Squeak Sword. The Red Brick costs 10,000,000.
Something else that could probably help is the Attract Studs extra - C7FJ7B.
TIP: A way to get to 10 billion faster is to try to avoid buying characters and other red bricks beside the multipliers until you reach 10 billion. Also, try to work your way to getting the 10x first so that after you spend the 10,000,000, you can regain the studs quicker since you'll have the biggest individual multiplier.
Thanks to SiebeTolsma for pointing out:
There seems to be a bug with this achievement that WB has even addressed: "Sometimes, the Achievement for collecting 10 billion studs (Delved too Greedily...) doesn't unlock once 10 billion studs are collected. If this occurs, you may need to collect 100 million more studs (10,100,000,000)."
In a way that makes sense since technically the achievement says to "collect more than 10,000,000,000", but that should just mean 10,000,000,001.
He also suggested that a good place to collect studs is in Minas Tirith right when you load up the game. Smash the tree in the center that will give you a whole mess of studs. Once you've collect all its studs, fast travel to the night version of the area and repeat this process.