To get more members for your alliance, other members have to apply for membership with you and your alliance. They just have to visit the profile page for your alliance for that and click on the button “apply"” to send you their request.
A little red digit will tell you, if another player has applied for membership with your alliance. You then simply open your alliance and click on the pen next to the player name to accept or decline membership. Via this pen you will also be able to change the status of a alliance-member after he or she has joined the club.
To get new members, simply write them a personal message and advertise for your alliance, so that they apply for membership with you. You can also ask others to join you using the global chat.