As far as it goes, that looks OK. A full schematic diagram would help.
If that is the full design, I would not expect it to work, you need additional decoupling and possibly osdcillator components before it will operate.
When you say the ULN2003 is not working, what exactly are you seeing? If it a transistor array so if you expect the output to go high when the input is low you are mistaken. It works like a voltage operated switch between the output pin and GND pin, when the input is high, the output is grounded, when the input is low the output is turned off. The output pin will not go high unless you provide it a current source from outside the device.
Brian.