Okay I see this alot in the forum, so I want to make a "how to" topic about it.
Firstly you need to get the level 10 space stage badge, the Omnipotent badge.
This makes it so when you first contact the Grox they will allow you to talk to them.
Now for relationship, dont destroy ANY of their ships, it will make them a tad bit more angry, but if you do by accident you can still ally with them, so no harm.
Remember to turn your autoturret off, and tell all your allied ships to leave, if you dont know how to do this...than you are a noob.
For the first part of the Grox, do at least five missions for them at request, this improves relations.
Then run around their planets using the Super Happy Beam, this makes them happy as well, and its possible to get a +20 or +30 (in some cases) relations boost with them if you use this "friendly tool".
A huge boost in relations for them involves blowing up planets nearby in neutral or alien controlled space(not the grox). This can give you a +50 boost if you do it all five times. I only did it three times, I think.
Now, also give them five gifts of 100,000. This gives you a +10, also helpful.
You also get a +10 for them meeting you as a gracious super power, I am not so sure what that is, but if you dont have it, you can still ally.
Once you reach the Blue smiley for them. Get 3 trade routes with 3 of their systems, this will get you a +30.
Now with this and other parts combined you should be able to get them into the green smiley, thus resulting in an alliance if you with.
Remember to never destroy their ships, steal their items, and anything else mean. They hate that alot.
Side effects of the alliance include the ENTIRE galaxy hating and going to war if you come into contact with any races. However, I rarely get attacked, and actually haven't gotten attacked at all so far, so possibly a fear AI factor.
Rewards include a nifty secret space badge,(The badge is Dance with the Devil.) and being allied with the most powerful race in the game, woot!
Also, the allied ship you get following you around is insanely buff.
Also, Gracious Greeter comes from being friendly in the tribal stage.
This gives a +10 for the gracious super power part, and being religious in the civ stage might help I think.
Some other notes.
Having a Grox ship in your fleet is like having the powers of God with you.
You can trade with the Grox and buy their planets, thus having the ability to construct Grox buildings when you buy the planet, since you will have a grox city now. You can build with non-ascessible items in a sense.
Its also possilbe to have even better relations with the Grox, supposedly if you have a very long alliance with them.
This hasnt been fully proved yet though.
One of the pics shows you my alliance points with the Grox, so you can get an idea on what to do.