If you plan on going through the chapters again in free play mode and/or getting 100% completion on this game, this achievement is something that you don't have to try at getting. I played through the chapters the first time, went straight for the red brick multipliers, and while going for the mini kits in free play mode had well over the required amount of studs by chapter 8. I did this while not even trying. Some of the chapters I was able to accumulate up to 3 billion studs, while ones that I breezed through I was still able to get 500 mil-1 billion. It just depends on how thorough you are in free play mode with destroying stuff and killing guys. So if you plan on replaying the missions, don't waste your time running around and chasing studs, as they will come. If you don't want to replay and just want the achievement, Pinicus did a good job of describing a method for just the achievement. To add to his, the walkthrough that was made for this game is a very good resource with locations for everything in this game. So if you don't want to watch lengthy videos and would rather have a map, see the walkthrough.
My recommendation for playing this game in the fastest way possible to 100% completion is:
1. Beat the story for the first time with little regard to mini kits and citizens in peril. In other words beat this as fast as you can.
2. Then I would unlock lex luthor, joker, and riddler so I could get the red bricks. They can be found on your map if you have the map clear by accessing bat terminals (red beams). Then go after the 2x, 4x, 6x, 8x, 10x multiplier red bricks quickly. Use super man to fly to the locations. Your last red brick with the multiplier should be on the bottom of the map. I got these right away so that I was able to make purchases as I worked my way up in step 3.
3. Then I would work my way back up the map getting everything in sight: characters, red and gold bricks, citizens in peril, and vehicles. Activate the batman terminals to view the whole map, and scan the map for locations of all these items. I used the red bricks (thanks to the cheat codes) for all the indicators so that they showed up on the screen and where easier to find. Just a quick note, you will not be able to get all the characters on the way up the map because you will not have enough gold bricks to build the arches.
4. After you have most of the items and worked your way back up the map, I would enter the batcave and do the free play on all chapters. Because you unlocked most of the characters, you will now have them and wont have to re-play the missions again because you didn't have someone. Like I said, about half way through this step or so you should get the 10,100,000,000 stud achievement.
5. Lego city play through, then mop up for missed characters on the map. You will need the last gold brick from lego city to get the last character, thats why its better in this order.
Bat terminals not active are red beams that shoot in the sky, blue beams means you have accessed them. These will open the map up and allow you to see everything when you scan.
Characters are unlocked by building arches and by going to their icons on the map. You can only build the arches by having the required amount of gold bricks. As far as the icons go, go there, fight the enemy, then you will have the option to purchase them by pushing B.
The hunt for gold bricks, red bricks, citizens in peril, characters, and etc is a grind. It was a long pain in the butt process. This will take you awhile, but is necessary for 100% completion
Some of the gold bricks require a batman or robin suit, and actually require you to climb a building via these suits. I found it easier to fly to the brick and backtrack on some of these to find the batman or robin signal with these suits because you will start a few buildings away. Plus if you backtrack you might be able to start mid way through the process by not having to climb buildings with say a magnet suit just to switch to the gymnast suit.
If you have any questions message me. Thanks to pinicus for his method, and to the makers of the walkthrough for their help. Before you post negatively let me know why and what I can do to make this better. I know that