How to get the ally to like the potential ally in Stellaris

There are a number of technologies you can research that increase the size of the borders around your outposts and colony ships, including

• Galactic Ambitions

• Manifest Destiny

• Propaganda Transmissions (Repeatable)

The Stellaris Wiki entry on borders also says that:

Border spread appears to work volumetrically. That is, as the empire becomes more dense and grows, its borders also appear to spill outward. Removing colonies or frontier outposts from within the empire's well-established borders may affect the outer border unpredictably.

So building additional colonies and outposts in your empire's interior can increase your borders at the fringes.

If you're expanding your empire via frontier outposts, it's worth noting that

Outposts controlled by sectors do not cost Influence to maintain.

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)