How can I use my Arduino together with an external power supply?
You connect the Arduino ground to the ground of your power supply and H-bridge driver. Make sure that the high currents have a direct ground path from the H-bridge back to the power supply and not through the Arduino's ground.

Figure 1. A simple driver circuit for H-bridge. Image source: Bristol Watch.

Since the Arduino can only switch up to 5 V it cannot &quot;reach high enough&quot; to control Q1 and Q3. For this reason the 4.7k pull-up resistors are added and the 2N2222 transistors (or any small NPN) are added to turn on Q1 and Q3.
Note that Q1 and Q3 are PMOS; Q2 and Q4 are NMOS. Your design shows four NMOS. You can do that but the switching circuit will be more complex.
Connect Arduino ground to H-bridge ground.