Instructions
Sign treaties with the other civilization -- active treaties improve your relations.
Research Xeno Ethics and select an ethical alignment for your civilization that agrees with the other civilization's alignment. Civilizations with identical alignments receive a relations boost.
Avoid displaying aggression toward the other civilization -- not just outright declarations of war, but the massing of ships near other civilizations' planets or the expansion of shared borders. Even if you're leaving alone the civilization you're trying to placate, you can still receive a relations hit for repeated acts of aggression against other civilizations, especially if they are friendly with the target civilization.
Increase your military strength. In the absence of aggression, a strong military gives other civilizations a relations boost toward you, just as a weaker one causes a relations hit.