How can I make other empires like me enough to join an alliance?

Solution 1:

So there are different methods of increasing relations with other empires:

• Establish an embassy: you can establish an embassy with the other empire in the same screen as the screen shot you provided. This will increase relations slowly until it reaches 100. You have a limit of 3 embassy by default but government type options can allow you extras.
• Rival the same empire: If the other empire has a rival, rivaling that empire will boost your relations by 25 ⁽¹⁾; this only lasts for the duration of the rivalry.
• Select xenophile traits: being a xenophile will increase other empire relations by 10, or by 20 if fanatic xenophile.

(1) Diplomacy - Stellaris Wiki: Opinion modifiers » Mutual rivals (17 May 2016)

Solution 2:

S.Wessels's answer hits several points, but note also the "relative power of empires" modifier. Other empires will be much more willing to enter in an alliance with you if you're stronger than them.

If you Guarantee the Independence of another empire, that will put you on the hook for some defensive wars involving them, but will boost your relations by 25. (You can make this less of a strategic loss by going for a mutual guarantee of independence, which is sort of like a defensive alliance.)