In Breaking Bad, Walt turns into a dangerous, egotistical, manipulative and most-wanted Man from a mild mannered, docile, family man.
He did start the whole thing for his family, but soon lost his way, which he admits himself during the series finale.
I want to know which is the first sign in the series where we see Walt's motives changing, where he was clearly not doing it for the family anymore.
TLDR:
First sign of "darkness": Season 2, Episode 6.
In it for himself: Sometime in Season 5.
Full Answer:
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 me out?
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.
Stay out of my territory.
I personally believe in this episode SE2EP10 Over, Walt completely embraced his dark side and chose to be Heisenberg rather than a family person.
IMDB plot:
Walt's cancer has greatly improved. Time to celebrate. Meanwhile Jesse tries to meet his new girlfriend's father.
Look at the facts, Walt's health condition is substantially improved and it is his ticket to say goodbye to Meth manufacturing and live with his family like a normal guy with his drug money. But during the celebration, he says,
It's kind of funny.
When I got my diagnosis, cancer, I said to myself,
Why me?
And then, the other day, when I got the good news...
I said the same thing.
The above speech by Walt clearly explains that he's not willing to let go of drug business and be happy with his family.
Walt has always been methodical and calculated of his actions but in the ending scene of the episode, at the supermarket, when he learns that there is competition for him in the drug business, his ego is hurt. The way he walks towards those guys (the competition) with this intimidating expression on his face and threatens them to stay out of his territory cleared things for me that he decided to stay Heisenberg rather than taking care of his family as priority.
I think it is pretty definitive by the time of Buyout (S. 5 Ep. 6). From the sub-titles:
Walter: Jesse, you asked me if I was in the meth business or the money business.
Neither.
I'm in the empire business.
I think the first sign was when Walt let Jane die because he wanted Jesse on his team to continue cooking and making money. I don't think Walt was ever in it for anyone but himself. His cancer was the catalyst for his transformation, but he was more than anything, getting revenge against the people he had always felt deep burning hatred for and getting out of the sheltered, Mr. Rogers life that he hated. The guy was clearly never fulfilled or happy, and was constantly in envy of the people above him. Case in point: Gray Matter. It was touched on a lot how he quit before they made it big, and then regretted his decision ever since. But instead of realizing that at any point, he could have called and gotten his job/lost livelihood back, he held onto his wounded pride and blamed them for his own mistakes/inaction.
I don't think it was that he was in it for his family and lost his way, I think it was that he was never in it for his family but was in denial about it and didn't get the balls to admit it until Felina when he finally accepted that he was a bad guy.
