Assuming you're using method of lines.
Let the original initial-boundary problem be
$$
u_t = u_{xx}\\
u(0, x) = f(x)\\
u(t, 0) = a(t)\\
u(t, 1) = b(t).
$$
Introduce a set of points $x_j = jh,\; j = 0,1,\dots, N,\;Nh = 1$.
Let $u_j(t) = u(t, x_j)$. Note that $u_0(t) = a(t),\; u_N(t) = b(t)$ are already known. Unknown are $u_j(t),\; j = 1, 2, \dots, N-1$.
Then $u_{xx}(t, x_j)$ can be approximated as
$$
u_{xx}(t, x_j) \approx \frac{u_{j-1}(t) - 2u_j(t) + u_{j+1}(t)}{h^2}.
$$
Plugging that into PDE gives
$$
u'_j(t) = \frac{u_{j-1}(t) - 2u_j(t) + u_{j+1}(t)}{h^2}
$$
a system of $N-1$ ODEs with initial conditions $u_j(0) = f(x_j)$. That could be solved using any RK method (provided that method is stable). For explicit Euler method that would be
$$
\frac{u_j(t_{n+1}) - u_j(t_n)}{\tau} = \frac{u_{j-1}(t_n) - 2u_j(t_n) + u_{j+1}(t_n)}{h^2}, \; j = 1, 2, \dots, N-1\\
u_j(0) = f(t_j)\\
u_0(t) = a(t), \; u_N(t) = b(t).
$$
This well known explicit scheme is stable when $\frac{\tau}{h^2} \leqslant \frac{1}{2}$.
Edit. For those who ask how to use this method with higher order RK methods. Let's take for example an RK2 method (explicit midpoint) with the following Butcher's tableau:
$$
\begin{array}{c|cc}
0 & 0 & 0\\
1/2 & 1/2 & 0\\
\hline
& 0 & 1
\end{array}
$$
Applied to ODE in form $\mathbf u' = \mathbf F(t, \mathbf u)$ this method expands to
$$
\mathbf r = \mathbf F(t_n, \mathbf u_n)\\
\mathbf s = \mathbf F\left(t_n + \frac{\tau}{2}, \mathbf u_n + \frac{\tau}{2} \mathbf r\right)\\
\frac{\mathbf u_{n+1} - \mathbf u_n}{\tau} = \mathbf s
$$
I've used $\mathbf r$ and $\mathbf s$ instead of common $\mathbf k_{1,2}$ to reduce the number of indices involved. Here $\mathbf r$ and $\mathbf s$ are intermediate values that depend solely on values of $u_j$ at $t = t_n$.
For our case the right hand side of the ODE is
$$
F_j(t, u_1, u_2, \dots, u_{N-1}) = \frac{1}{h^2}\begin{cases}
u_0(t) - 2 u_1 + u_2, &j = 1\\
u_{j-1} - 2 u_j + u_{j+1}, &1 < j < N-1\\
u_{N-2} - 2 u_{N-1} + u_N(t), &j = N-1
\end{cases}
$$
Note that $u_0(t)$ and $u_N(t)$ are given explicitly as $a(t)$ and $b(t)$. This is why I have separated cases $j=1$ and $j = N-1$ in the definition of $F_j$.
Putting this altogether gives
$$
r_j = \frac{u_{j-1}(t_n) - 2 u_j(t_n) + u_{j-1}(t_n)}{h^2}\\
s_j = \frac{1}{h^2}\begin{cases}
u_0\left(t + \frac{\tau}{2}\right) - 2 \left(u_{1}(t_n) + \frac{\tau}{2}r_{1}\right) + \left(u_2(t_n) + \frac{\tau}{2}r_2\right), &j = 1\\
\left(u_{j-1}(t_n) + \frac{\tau}{2}r_{j-1}\right) - 2 \left(u_j(t_n) + \frac{\tau}{2}r_j\right) + \left(u_{j+1}(t_n) + \frac{\tau}{2}r_{j+1}\right), &1 < j < N-1\\
\left(u_{N-2}(t_n) + \frac{\tau}{2}r_{N-2}\right) - 2 \left(u_{N-1}(t_n) + \frac{\tau}{2}r_{N-1}\right) + u_N\left(t + \frac{\tau}{2}\right), &j = N-1
\end{cases}\\
\frac{u_j(t_{n+1}) - u_j(t_n)}{\tau} = s_j
$$
Note that $r_j$ and $s_j$ are helper values to step from $u_j(t_n)$ to $u_j(t_{n+1})$ and are different for each time step. One may want to attribute values $r_j$ and $s_j$ to some moment of time. A reasonable choice would be attributing all each value $\mathbf k_i$ with moment $t_n + \tau c_i$. Here $c_i$ is the first column of the Butcher's tableau.
$$
r_j(t_n) = \frac{u_{j-1}(t_n) - 2 u_j(t_n) + u_{j-1}(t_n)}{h^2}\\
s_j\left(t_n + \frac{\tau}{2}\right) = \frac{1}{h^2} \times \\ \times \begin{cases}
u_0\left(t + \frac{\tau}{2}\right) - 2 \left(u_{1}(t_n) + \frac{\tau}{2}r_{1}(t_n)\right) + \left(u_2(t_n) + \frac{\tau}{2}r_2(t_n)\right), &j = 1\\
\left(u_{j-1}(t_n) + \frac{\tau}{2}r_{j-1}(t_n)\right) - 2 \left(u_j(t_n) + \frac{\tau}{2}r_j(t_n)\right) + \left(u_{j+1}(t_n) + \frac{\tau}{2}r_{j+1}(t_n)\right), &1 < j < N-1\\
\left(u_{N-2}(t_n) + \frac{\tau}{2}r_{N-2}(t_n)\right) - 2 \left(u_{N-1}(t_n) + \frac{\tau}{2}r_{N-1}(t_n)\right) + u_N\left(t + \frac{\tau}{2}\right), &j = N-1
\end{cases}\\
\frac{u_j(t_{n+1}) - u_j(t_n)}{\tau} = s_j\left(t_n + \frac{\tau}{2}\right)
$$
While this is the answer to the question "How to apply RK2 to this ODE" I really don't like the final form. Instead let's write the same method in a slightly different form:
$$
\frac{\mathbf u_{n+1/2} - \mathbf u_n}{\tau / 2} = \mathbf F(t_n, \mathbf u_n)\\
\frac{\mathbf u_{n+1} - \mathbf u_n}{\tau} = \mathbf F(t_n + \frac{\tau}{2}, \mathbf u_{n+1/2}).
$$
One can check that the methods are the same by plugging $\mathbf u_{n+1/2} = \mathbf u_n + \frac{\tau}{2} \mathbf r$. Just like $\mathbf r$ and $\mathbf s$ the $\mathbf u_{n+1/2}$ is a helper value used to perform a step from $\mathbf u_n$ to $\mathbf u_{n+1}$.
Applied to our ODE this method gives
$$
\frac{u_j(t_{n+1/2}) - u_j(t_n)}{\tau / 2} = \frac{u_{j-1}(t_n) - 2 u_j(t_n) + u_{j-1}(t_n)}{h^2}, \quad j = 1, 2, \dots, N-1\\
u_0(t_{n+1/2}) = a\left(t_n + \frac{\tau}{2}\right), \quad
u_N(t_{n+1/2}) = b\left(t_n + \frac{\tau}{2}\right),\\
\frac{u_j(t_{n+1}) - u_j(t_n)}{\tau / 2} = \frac{u_{j-1}(t_{n+1/2}) - 2 u_j(t_{n+1/2}) + u_{j-1}(t_{n+1/2})}{h^2}, \quad j = 1, 2, \dots, N-1\\
u_0(t_{n+1}) = a\left(t_n + \tau\right), \quad
u_N(t_{n+1}) = b\left(t_n + \tau\right).
$$
Though not strictly necessary I have defined values $u_0(t_{n+1/2})$ and $u_N(t_{n+1/2})$ to get rid of treating $j=1$ and $j=N-1$ as separate cases.
Best Answer
I have programmed your differential system using one of the Runge-Kutta "blackboxes" of Matlab. Here is the result :
Fig. 1 : Case $\mu=0.1$. Superposition of the vector field and "trajectories" ("pathes" as you call them) with initial points of the form $(x_0;0)$, that indeed follow the direction indicated One sees a spiral effect when $|x_0|<1.$
How can be explained this spiral effect when $x$ is small ?
This is because system :
$$\begin{cases}\dot x = μx − y + xy^2\\ \dot y = x + μy + y^3\end{cases}$$
can be written :
$$\underbrace{\begin{pmatrix}\dot x\\ \dot y \end{pmatrix}=\begin{pmatrix}\mu & - 1\\ 1 & \mu \end{pmatrix}\begin{pmatrix}x\\ y \end{pmatrix}}_{(S)}+y^2\begin{pmatrix}x\\ y \end{pmatrix}$$
which appears, in the vicinity of the origin ($x,y$ "small enough"), as a little perturbation of system (S). This system is classical and leads to a spiraling behavior. Did you know it ? It is very similar to Lotka-Volterra equations (https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations).