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Heisenberg's Equation of Motion states that a Matrix's time evolution is determined by the following: $$\frac{d\hat A}{dt}=\frac{1}{i\hbar}\left[\hat A,\mathcal{\hat H}\right]$$ Where the Matrix $\hat A$ is a quantum operator and $\mathcal {\hat H}$ is the particle's Hamiltonian.

Heisenberg postulated that the relations between Quantum Mechanical operators should have the same form as their classical equivalents, namely Hamilton's Equations of Motion, in this case for a particle moving in one dimension $$\dot x=\dfrac{\partial \mathcal H}{\partial p}$$ $$\dot p=-\dfrac{\partial \mathcal H}{\partial x}$$ So if our Hamiltonial is composed of the Kinetic energy term $\frac{p^2}{2m}$ and a potential energy term which is represented as a power series of x, we have $$\mathcal H=\frac{p^2}{2m}+(a_1x+a_2x^2+a_3x^3+...)$$ so $$\dfrac{\partial \mathcal H}{\partial p}=\frac{p}{m}$$ $$\dfrac{\partial \mathcal H}{\partial x}=a_1+a_2x+a_3x^2+...$$ So we have' $$\dot x=\frac{p}{m}$$ $$\dot p=-(a_1+a_2x+a_3x^2+...)$$ So, with Heisenberg's Postulate: $$\dfrac{d\hat x}{dt}=\frac{1}{i\hbar}\left[\hat x,\mathcal{\hat H}\right]=\frac{\hat p}{m}$$ and $$\dfrac{d\hat p}{dt}=\frac{1}{i\hbar}\left[\hat p,\mathcal{\hat H}\right]=-(a_1+a_2\hat x+a_3\hat x^2+...)$$ How can I solve these to obtain $$\hat x\hat p-\hat p\hat x=i\hbar I$$

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Heisenberg's equation of motion cannot derive the canonical commutation relationship. The latter must be postulated further to Heisenberg's equation. The reason is simply that Heisenberg's equation is an equation of motion - a differential equation - and one needs initial conditions to fully define an observable for all time. Thus, a canonical commutation relationship is encoded into the initial conditions, and the best that you can do with an equation of motion is to prove that initial relationship is time invariant.

Let's look at your problem in detail.

I'm not 100% sure of the history here, but the fundamental idea (or at least, I believe, how Dirac saw it) in quantizing a theory is to replace Possion brackets by Lie brackets in the theory's Hamiltonian formulation - with the multiplying constant $1/(i\,\hbar)$.

The main outcome of such a procedure is Heisenberg's equation of motion from the Lioville equation, of which a special case is Hamilton's equations for conjugate variables. The replacing of the Poisson brackets in all of these, when applied to the Hamiltonian you have, indeed leads to all your equations. So now apply them to $[\hat{x},\,\hat{p}]$ and you duly find:

$$\mathrm{d}_t [\hat{x},\,\hat{p}] = 0$$

which is indeed consistent with the CCR. But you need to know the initial value of $[\hat{x},\,\hat{p}]$ to pull off what you seek to do. It should be clear that this initial condition is needfully information that is not contained in what you have postulated.

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