You can get this achievement through one of two ways:
1.) You play naturally, and let progression take its course. Upgrading machines is a vital part of gameplay, and this achievement is cumulative, so it is one of those achievements that you just get Eventually™ depending on your play style. It is worth noting that you don't get the ability to upgrade machines until the second hospital, so that is when progress officially starts. Janitors with the necessary skills are required, and will not upgrade automatically. In order to do so, you need to select the machine you want to upgrade and choose that option. The game will also pester you about upgrades if you aren't.
2.) You game the system. A lot of the achievements in this can be "gamed" in straightforward ways. In most cases, achievements can be unlocked through doing the same missions repeatedly. In the case of this achievement, you don't even need to do that. Assuming you have upgrades unlocked, start a mission and ignore the requirements. Instead, build the rooms with machines that can be upgraded only. Hire a janitor that can do it, and have them upgrade the machine. Once it is done, sell the machine and buy another one. Repeat as necessary. You could also buy multiple copies of the same room, hire multiple janitors, and have them all upgrade at once to speed up the process.
Once you hit 25, the achievement should unlock in the level.