This is a tricky achievement, as you'll need a sustainable and well funded hospital to even sniff Level 20. I earned this on Smogley, which required a Level 16 for Three Stars anyway, so it was a case of pumping it up 4 extra levels.
Increasing your hospital size, staff and upgrading machinery are the multiple ways to increase your Level. This will entail the following:
Size: As you grow and earn money, you can buy extra plots of land. I only recommend you do this once you have a consistent fund of money, and there is a bit of a loophole for this. If you set up a Marketing room, and put $130,000 dollars into a large campaign, you'll be able to jack prices up to 100% (to do this, press cn_X, cn_RB twice, highlight Prices and then tap cn_RB until everything is maxed. This means you'll get tons of money while not suffering a REP penalty, provided you simply keep doing a Large REP marketing campaign. You'll be rolling in dough before you know it.
Staff: Hiring more staff will add Hospital points, and upgrading them further will too. Make sure you promote them once they email you requesting a Promotion! This adds points too, though it's not as much as hiring staff. Note you can also hire more staff depending on your Hospital level, so keep checking back. But I believe after Level 15, the staff limit is no longer increased, meaning if you fill your roster, you'll need to rely on upgrading and promoting your Staff. Build a nice, big training room.
Machinery: Creating rooms with machines will add a lot of points towards your Level, and you can then hire Mechanics-trained Janitors to upgrade. Note, you will need to fully research machine upgrades in order to order them. All machines can be upgraded to Level 3, so if it is not letting you upgrade, check your Research station! Room Prestige also adds points to your level, so I recommend you plaster the walls with Gold Star Awards in order to bump the room prestige up to Level 5.
It's not an easy achievement, and will take a lot of time in a single hospital - most likely beyond earning 3 Stars - but hopefully these tips will steer you in the right direction!