To unlock this achievement, you must unlock all locks on the sphere grid, fill all empty nodes, and active all nodes with one character. There is also an achievement for doing this with all characters.
To begin, I recommend fighting One-eye until you get a weapon drop with the "Triple AP" ability and 2 free spaces for each character you wish to boost. Once you have your "Triple AP" weapon, add "Overdrive -> AP" and "Triple Overdrive" to the weapons. If you are having trouble killing one-eye, take a break to capture more/all of the required fiends for the monster arena, then use the Magus sisters to fight him. By this time, their attacks should do near 99,999 damage each and you should kill him in only a few hits.
Triple Overdrive requires 30x winning formula. You get 99x for capturing 1x ever fiend in every area. You can obtain more by bribing Sand Worms (Bikanel) to get 15x Winning Formula for 900,000gil. If you don't want to get those, Double Overdrive works as well and can be purchased for 30x Underdog Secrets. 99x of these can be obtained by speaking to Rin on the airship after collecting all Al Bhed Primers, or you can get more by bribing the chimera from Home and get 15x for 180,000gil.
Overdrive -> AP requires 10x Door to Tomorrow. You get 99x Door to Tomorrow for capturing one of every fiend in 6 different areas.
Once you have at least 3 weapons completed, you will need to fight the Area Conquest Fiend, Don Tonberry, which is unlocked by capturing 1x each monster in the cavern of the stolen fayth. He has a special attack, called Karma, which is a counter-attack that does damage based on how many fiends each character has killed. For this fight, it is incredibly helpful to have 1-2 people with the "Auto-phoenix" ability on their armor, to greatly reduce time to revive KO'd characters. Before you begin the fight, set each character's overdrive mode to "Comrade" (note- if one character is taking significantly more damage, set his/hers to "stoic" instead) and begin fighting Don Tonberry. Attack him until he gets close enough to stop counter-attacking, then flee. You should get near 99x sphere levels for all 3 of your characters.
When you inevitably run out of power, speed, and magic spheres, there is a very nice trick to re-stock them quickly. Kottos (Area Conquest, 1x each fiend in Mi'ihen highroad) drops 20x healing springs for a regular kill or 40x for an overkill. Simply use the ability or item to distill power/speed/magic/ability spheres to quickly restock. Be sure to use an aeon for an overkill if you are unable to do so with your regular attacks. Remember to swap your AP boosting weapons for break damage limit weapons before the fight if you have them.
To get the empty nodes filled, you will need to kill fiends in the calm lands fiend arena Species Conquest list. The following fiends each drop the attribute spheres, 1 item for a regular kill or 2 for an overkill:
1. Fenrir - Agility Sphere
2. Pteryx - Evasion Sphere
3. Hornet - Accuracy Sphere
4. Vidatu - MP Sphere
5. One-eye - Magic Defence Sphere
6. Jumbo Flan - Magic Sphere
7. Tanket - Defence Sphere
8. Juggernaut - Strength Sphere
9. Ironclad - HP Sphere
I recommend focusing on 3 players when leveling in this fashion rather than trying to keep all 7 around the same stats. It also helps if you use a teleport/return/friend/warp sphere to get the 3 characters in the same area when you begin filling in the empty nodes. Once you get to a point where your characters are hitting 99,999 damage regularly with their break damage limit weapons, it is much easier to farm the attribute spheres needed. When I started filling in, I had 99x Magic Defence spheres from killing one-eye for my weapons and used those until I was able to consistently hit 99,999 and begin farming the others. Also, as I said earlier, the Magus Sisters are an enormous help early on, as their special attacks can easily overkill these fiends much earlier than your characters will be able to.
If you have any questions, please ask in the comments below. If you negative vote, please let me know why so I can fix the issues!