Declare war without involving the allies

You occupy the planet, you do not control it, this works similar to EU4.

The occupation grants a large amount of warscore and you can send your armies away, Your opponent has to reinvade to liberate the planet.

Whether you get the planet in the end depends on the war goals, the occupation merely provides additional warscore, with which you can press your demands.

Most importantly though, it prevents the enemy from using the planet and getting resources from it without you having to constantly bombard it and requiring an invasion from your opponent to liberate it.

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)