Working through Murdock's Pertubations [1.5.5 and] 1.6.3 here...
We're working to solve $x^3 - 2x^2 + x - \epsilon^2 = 0$.
When trying with rescaled coordinates, I'm having trouble choosing coordinates that don't have multiple roots. I tried $y=x/\epsilon$ and $y=x-1$, but I'm still stuck with double roots. What should I be trying instead?
The best thing to do is to sketch the unperturbed situation, i.e. the graph of $x^3 -2 x^2 +x$. The roots of $x^3 - 2x^2 +x$ are the intersections of the graph with the horizontal axis. When $\epsilon > 0$, this graph is shifted slightly downwards. As you can see, the root at $x=0$ shifts slightly to the right, whereas the double root at $x=1$ splits into two roots, both close to and on either side of $x=1$.
For the root near the origin, you tried $x = \epsilon y$. When you substitute that in the equation and expand in $\epsilon$, the leading order term implies $y=0$ -- but then you're back where you started, since you already knew that the root you're looking for is near $x=0$. To determine what goes wrong here, I suggest you study example 1.5.2 in Murdock.
For the two roots near the double root at $x=1$, you tried $x = 1+y$. When you substitute that in the equation and expand in $\epsilon$, the leading order term again implies $y=0$, so again, you're back where you started. Again, something goes wrong, but now for a different reason! To determine what goes wrong here, I suggest you study example 1.5.3 in Murdock.