To join an Alliance, tap on the "Social" icon on the bottom right corner, then head to the “Alliances” tab. From here, tap on the “Join an Alliance” drop-down arrow to view pre-existing Alliances or search for a specific one if you know the name. 

When you locate the Alliance you wish to join, tap on the banner and then select the “Request to Join” button. Please note, some Alliances will only approve members by request or cannot accept new members if they recently reached the maxmimum amount fo 50 members.