I made a custom ATmega32u4 board for my project. Components came from clone of Arduino Pro Micro (with bootloader and code which I uploaded) from AliExpress. I soldered everything in to my board according to my schematic but the board doesn't work. I don't have any signal from that.
You have an error in your schematic.
You have used the net name VCC for your main power circuit. However your USB incoming power is labelled as +5V.
That means you have two separate nets - +5V with power on it, and VCC isolated from it entirely.
You need to connect those two nets together - name them both VCC or both +5V so they form a single net (easiest is to adopt VCC since you use that on other connections too).
To "fix" your current PCB you could connect a small wire from the cathode of D2 to pin 44 AVCC on the main chip which will bridge the two nets together.