Health can be raised by praying at shrines and eating fruit, while stamina can be raised by eating silver lizard tails or killing colossi.
I had read that it would take 5-7 playthroughs to get this, but nobody had a definitive number. Here's what I wound up doing.
Full playthrough on Easy to learn the game
Normal NG+ Time Attack (doesn't add to health or stamina)
Hard NG+ Time Attack (doesn't add to health or stamina)
Speedrun full playthrough on Hard NG+ (with Queen's Sword/parachute/invisibility cloak) for
After that, my health was about half full, and my stamina 1/3. I reloaded my Hard save after the 15th colossus, and ran around collecting fruit. I wound up eating 75 fruit total across both story saves before health stopped increasing. Silver-tailed Lizards didn't seem to increase stamina much at all (one colossi is worth ~5 lizards), so I only ate 3 total, 2 of which were for separate achievements.
I went with a strategy of starting a NG+ on Easy, killing the first two colossi with the Queen's Sword in a little under 5 minutes, then restarting a NG+ from that save. When the trophy popped, I checked my stats, and had exactly 23 non-story/non-time-attack kills each on colossi #1 and #2, so after two story-mode runs, you'll need 46 colossi killed. If you prefer to play through the whole story, it should pop around #14 on your 5th playthrough, but this took me a hair over 2 hours, which was just shy of my speed run time for a single playthrough, so it seems much faster, if a little boring.