During the game, you will come across machines called Gatherers Gardens. Here, you are able to buy Tonic slot upgrades for 80 ADAM and Plasmid slot upgrades for 100 ADAM.
For this achievement you will need a single slot so the minimum you will have to spend is 80 ADAM.
A way of earning more ADAM is to harvest little sisters but this will mean that you will miss out on the achievement for rescuing all the little sisters. If you do rescue every little sister, you will have enough ADAM to buy all of the upgrades by the end of the game.
The Maxed One Track achievement in BioShock worth 24 pointsPurchased every slot in one of the Plasmid or Tonic tracks