First sign of "darkness": Season 2, Episode 6.
In it for himself: Sometime in Season 5.
In my own opinion, the first glimpse I had of Walt's true nature was way back in Episode 6 of Season 2 - Peekaboo.
In this episode, Gretchen Schwartz visits the White household and is thanked by Skyler for giving the money to Walt for his cancer treatment. However, Walt had turned Gretchen down, so this leaves her confused and asking where he got the money from.
Gretchen arranges to meet Walt at a restaurant. She asks him why he lied and how he's paying the huge sums of money his cancer treatment costs. He tells her he owes her an apology - but not an explanation. She doesn't like his answer and the following dialogue occurs:
Gretchen: ...this isn't you.
Walt: What would you know about me, Gretchen? What would your
presumption about me be, exactly? That I should go begging for your
charity? And you waving your chequebook around like some magic wand is
gonna make me forget how you and Elliott-- How you and Elliott cut
Gretchen: What? That can't be how you see it.
Walt: It was my hard work, my research, and you and Elliott make
millions off it.
Gretchen: That cannot be how you see it.
Walt: Good. That's beautifully done.
Gretchen: -You left--
Walt: You are always the picture of innocence.
Gretchen: -You left me.
Walt: -The picture of innocence. Just sweetness and light.
Gretchen: You left me. Newport, 4th of July weekend. You and my father
and my brothers and I go up to our room and you're packing your bags,
barely talking. What? Did I dream all of that?
Walt: That's your excuse to build your little empire on my work?
Gretchen: How can you say that to me? You walked away. You abandoned
us, me, Elliott.
Walt: Little rich girl just adding to your millions.
Gretchen: I don't even know what to say to you. I don't even know
where to begin. I feel so sorry for you, Walt.
Walt: Fuck you.
To me, this was the first revelation on the show of the deep anger Walt had towards the Schwartzs and how bitter he was at losing out on the billions they now had.
At the time, it was the darkest we'd ever seen Walt on the show and certainly showed a new side of him.
As the series progresses, we continue to see this bitterness over Gray Matter Technologies, e.g. in the episode Buyout, when Walt reveals to Jessie the company is now worth over 2 billion dollars.
Therefore, to me this was the first sign that Walt wasn't just doing it for his family any more - that getting money was his own way of making something of himself given what he had lost by walking away from Grey Matter Technologies.
However, your question specifically asks "Which is the first sign that indicates that Walt is not doing it for the family anymore?" - I'm not sure this question can really be answered. Right up until the end of the show, in his own way, he's doing this for his family. He's doing it for him too, primarily him - but also for his family.
For example, after the exchange above, in Season 2, episode 9 (4 Days Out), Walt thinks his cancer is getting worse and goes into meth-cooking overdrive to ensure he has as much as possible for his family.
Throughout Season 3 we see Walt slowly barge back into Skyler's life because he doesn't want to be away from her, or them. In Season 3, episode 10 (Fly) we see Walt lose himself trying to kill a fly in his meth lab - because any contamination whatsoever could be catastrophic for them and their lives, including their families. Walt is fearful he can't protect Jessie or his family.
In Season 4 we see Walt plan the murder of Gus as Gus threatens both him and his family. He does this out of fear for his own life, yes, but also out of fear for his family.
In Season 5, Walt originally begins cooking again because Skyler gave almost all the money away to Ted. He has almost nothing left, so he cooks again for the same reason he did in Season 1 - to provide for his family. I feel it's only during this season, when the huge bucks come in, that he truly loses sight of his overall goal of providing for his family. For example, in Season 1 Walt states he needs $737,000 to provide for his family (worked out by calculating college fees, mortgage payments etc). At one point in Season 5 he has over $80 million.
So in summary:
His true nature is revealed in Season 2, Episode 6 in my opinion. However, although we see this "dark" side of him, he continues to be in it primarily for his family right up until Season 5 when, at some point, he loses focus and gets lost in his own reputation and the huge sums of money he is making.