“How Hard Do I Need to Press the Strings?”
Enough pressure so the note rings out.
That’s it.
Practice this by slowly applying pressure to the string and stopping once the note sustains.
It’s probably way less force than you were doing.
You really don’t need to death grip the guitar neck. If you want to make your life easier, take your guitar to a local music store or guitar tech and have them “set it up”. This will make the guitar play in tune and lower the string height. When your guitar is easier to play, you’ll help eliminate unnecessary tension to compensate.
Save your frets and hand pain with this one simple trick.