In addition to achieving "Close" relations, you also need to research the "Alliances" tech (available on the "Interstellar Governments" branch) in order to make an alliance with another civilization.
Another relations factor that is harder (but more fun!) to achieve is building a large fleet. I have found that if your fleet has relative parity with another civilization, it won't appear in their factors list; if your fleet has an advantage, it will appear as a single "+"; and if your fleet has a significant advantage, it will appear as a "++" factor. Basically, the bigger your fleet, the friendlier the other civilizations are toward you. (Who wants to make an alliance with a civilization that fields a wimpy fleet, anyway?) In fact, fleet strength should be one of the first considerations when deciding to make an alliance. If they have a week fleet, chances are you are going to get dragged into a fighting a war for them when one of the stronger civilizations goes after their planets and/or resources. On that note, make an alliance with a minor civilization only if it is absolutely critical that they be defended -- for instance, if their location is very strategic or close to your planets and/or resources. In any case, be careful with alliances as they are difficult to break with significant consequences.