Can you explain me what part of high school mathematics I should master before diving into differential calculus?
It depends on your current strength and weaknesses and also interests. If you tell me about them, I am able to give you a better answer. 
In general, you can interpret the high school mathematics as a toolbox for the higher mathematics, and there are a few parts that always be helpful.

1. Algebra: As simple as it is, it provides the basis for any other field of mathematics. Get familiar with roots and logarithms and exponential if you know the basics, and how to use them. 
2. Trigonometry is needed everywhere, if you feel comfortable with the angle theorems, you are not distracted when solving equations. 
3. The concept of functions is a wide field itself and is of great importance for solving differential equations. 

Most importantly, train to solve mathematical equations or puzzles, if you enjoy having difficult problems to solve, you will improve your skills
That seems like a lot of work. Do you think you can provide a means for it to be a lot earsier, maybe a study guide.