How can I connect a thin relay to a 220V wire?
The relay is designed to be soldered to a PCB. You can solder wires directly to it, but it will not be  a strong connection. You could try soldering it to a scrap of Veroboard (stripboard) and then connecting the wires to the board.

Probably best not to use those uninsulated crocodile clips (or are they alligator clips?) with 220V.

The best way is to get a bigger relay with bigger terminals. It's possible to buy relays which plug in to a base or socket. The base then has screw terminals for the wires.

Do not use a solderless breadboard for making 220V AC connections like this.