Within your city walls, you need to build an embassy. A level 1 embassy enables you to join an alliance. Once the embassy is complete, on the top part of the game, click the "alliance" tab. Within the alliance tab, you will see one that says "Alliance List". Click it and search the list of alliances. Click "message" next to the leader of the Alliance you want to join and wait for an answer. Or an alliance leader can invite you to the alliance.