According to the Wikipedia page on torus knots a (p, q)-torus knot can be given by the parametrization:
$\begin{cases} x = \cos{pt}\cos{qt} + 2\cos{pt}\\ y = \sin{pt}\cos{qt} + 2\sin{pt}\\ z = -\sin{qt} \end{cases}$
where $t \in \left[0, 2\pi\right]$.
Unfortunately, on its own a parametrization isn't very useful I think, aside from allowing one to graph the knot in something like Mathematica. That's because a knot is an embedding of $S^1$ in $\mathbb{R}^3$, not $\left[0, 2\pi\right]$ in $\mathbb{R}^3$.
My question is can someone walk me through the process of converting a parametrization of a torus knot call it $f: \left[0, 2\pi\right] \rightarrow \mathbb{R}^3$ into an explicit embedding $g: S^1 \rightarrow \mathbb{R}^3$? More generally, is there a process for converting a parametrization of a knot into an explicit embedding of $S^1$ in $\mathbb{R}^3$?
Sure. $S^1$ actually is $[0, 2\pi] / \sim$, where $a \sim b$ iff $b - a$ is a multiple of $2\pi$.
Since the map $X$ defined by
$$X : [0, 2\pi] \to \Bbb R^3:t \mapsto (x(t), y(t), z(t))$$
satisfies $X(2\pi) = X(0)$, it passes to the quotient, $S^1$, and thus is an explicit map from $S^1$ to $\Bbb R^3$. It's not hard to check that it's an embedding, because on the quotient, it's 1-1.
Or did you have some other definition of $S^1$ that you prefer?