Show that V is not a vector space under vector addition and scalar multiplication

It's a little easier to see if we look at the problem using matrix multiplication. Consider that $2x+y-z=1$ could be written $pmatrix{2 &1 & -1}^Tpmatrix{x & y & z}=1$. Now we can see that for some scalar $lambda neq 1$ we have that $pmatrix{2 &1 & -1}^T lambda pmatrix{x & y & z}=lambda neq 1$ so the space is not closed under scalar multiplication. Since $v+v=2v$ this also shows the space is not closed under vector addition.

I'm not quite sure where to start with this.

Whenever you're trying to establish that some set doesn't have a particular structure, it's useful to start with trying to find things that result in it failing to have that structure.

More specifically, if you want to show that your given set, under the desired operations, isn't a vector space, then you just need to show that it doesn't satisfy one of the vector space axioms. That's it.

So, we can see that $(0,0,0) notin V$. Otherwise:

$$2 cdot 0-0+0 = 1$$

$$implies 0 = 1$$

which is clearly false. This automatically prevents $V$ from being a vector space.