I think that at the beginning the best way to play the first 30 exercises of Hanon is to concentrate on relaxation.
For example, if you are playing exercise 1, you should play the first note, focus on whether you have the rest of the fingers, wrist, shoulder, etc. relaxed, then play the second note and so on. Be patient. It is very important.
After a while you can increase speed. If you focus on relax you will notice you can use some little wrist rotation movements to play this exercise, and have a free and fast hand. The same for the rest of exercises.
I use to play these exercises with the fingers just on the notes before pressing them (not lifting fingers or very little). If in some works (Baroque for example) I need an articulate style I lift a little bit more the fingers. It is not necessary to lift high the fingers to have a louder sound, so you should not follow Hanon advice IMHO. You would have a poor, harsh sound.
Instead of looking for speed try to focus on playing absolutely free and relaxed. Speed will come if you do so. If you play fast and tense, even if it is clean and articulated, may be you are doing a bad job .
For the rest of the exercises, each one has a different technique (octaves, double thirds, etc.)