To establish an alliance with a player, first you must research the appropriate knowledge, helpfully named 'Alliances'
Having the knowledge however is not enough. You will not even be able to propose an alliance with an AI until the relationship meter on the "Foreign Relations" tab of governing is "Close". This requires having lots of green + modifiers for the other side, and very few, if none at all red - modifiers.
A few hints on tips on getting the Ally option to appear:
- Only have 1 or 2 opponent AIs
- Have all the opponent AIs match your side (ie, if you are playing a Kingdom, do not have any Empire factions in the game)
- Establish trade routes (caravans) as soon as you can
- Do not build cities too close (negatives for competition for same resources) but also don't be too far away (negatives for being outside sphere of influence).
- Do not take actions to progress other victories (especially don't build any of the towers for the spell of making, or declare war on any of the players)
- Have a strong military. No, you aren't going to use it for conquering, but there are negatives for 'your military is weak' and positives for 'our military is weak'
- Do not accept tribute requests! Even if the AI offers you a tribute, it shows up in diplomacy as 'You are forcing us to pay tribute'
- Avoid AIs with personalities like 'unstable' or 'paranoid'
As soon as the relationship changes to close, propose an alliance. You may have to throw a gold piece or two at them to get them to accept. Alliances last for 99 turns, so don't think you have to propose them all at once - grab them as you can.