How can I copy the running-config file from a Cisco 4503 switch to a tftp server, which is on my PC? I connected the PC to the switch via the management port. I tried to issue "copy running-config tftp://192.168.0.1", but it didn't work.
You must be able to ping the pc from the switch. Just connecting it into the management port isn't necessarily going to make it work. Easiest thing would be simply to connect the switch to a normal port on the 4500 and assign it to one of the vlans. Give it an IP address out of that vlan and then as long as the 4500 has a vlan interface for that vlan you should be able to ping the PC. Then try the tftp.
Note, just in case, also check you don't have a personal firewall/anti-virus on your laptop which could be blocking the download.