It is highly advised the player doesn't do anything that will downgrade the Grox's relations, as their relations with them will change for the worse much faster than other empires, and a peace settlement isn't available to bribe them once the player go to war.
While attempting to ally with the Grox, there is a very important thing to remember about them: Many conventional boosts are much less effective on the Grox than any other empires. For example, giving a gift of §100,000 will only give the player +2 relationship instead of +10, and breaking the Galactic Code actually increases relations with the player.
Breaking the Galactic Code and keeping other empires calm is easy, and can considerably boost the relationship with the Grox. Buy a Planet Buster and fly into Grox territory. Once there, the player will need to get at least 8 parsecs away from the nearest empire, as they will most likely declare war on the player if doing it too close. Fly into a Grox system and blow up one of the planets that is not inhabited. Do not blow up a Grox planet; their relationship will go down even further. Doing this a few times, and the player will get a 50+ on their relationship status. The player can also just use the Scientist's Gravitation Wave or Zealot's Fanatical Frenzy ship tools on a non-Grox empire a few times within 8 parsecs of a Grox system.
Having an alliance with the Grox comes with both good and bad benefits. The good benefits are that the Grox will not attack the player, they will send one of their ships to join the player's fleet (3600 health), the player can use Fanatical Frenzy on them without any Grox retaliation, and the player can get to the galactic core without being attacked by them. The most prominent drawback is that nearly every empire's relations in the galaxy will reduce tremendously.