So.. form what I understand:

- Gray Fox was good.
- in MG2 he defects to Big Boss (aka Snake)

Questions:

1) Why does he help solid snake during MG2?
2) Why does he help Solid Snake during Metal Gear Solid even though Solid Snake killed him in MG2?

1) No idea, I never played the original MGs
2) From what I understand, Fox oringinally was helping Snake out to ensure that he would survive long enough for a fight-to-the-death, but by the end of the game, it's because (as Snake says after the Fox boss fight) the two of them were friends. Snake killed Fox in MG2, but only because they were soldiers on opposing sides. When they were away from the battlefield, the 2 never stopped being friends and seeing as fighting is their lives, they don't see it as a reason to ruin that friendship.

I don't think they were friends as such, it was more like Gray Fox saw that Solid Snake was a worthy opponent, so sort of a cross sided friendship. I think.

Out of respect for Snake, plus he also owed him for "saving" him in Outer Heaven. Plus I believe Big Boss ordered him to.

Fox knows that Snake "killing" him was just business and Snake had no ill will towards him, in the same way Fox had no ill will towards Snake. Snake explains it in MGS:

Respect is the reason Fox helps him. Plus he wants Snake to tell Naomi about him killing her parents.

Also, before MG2 they were friends. Or at least not enemies. Like Helikaon says, in MG2 they are on opposite sides, but it's their job.