Method 1:
Point $A$ and point $B$ ($A \ne B$) determine a line. You can find its equation. See if the coordinates of point C fits the equation. If so, A B and C are colinear, or else, no.
Method 2:
Point $A$, $B$ and $C$ determine two vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$. Suppose the latter isn't zero vector, see if there is a constant $\lambda$ that allows $\overrightarrow{AB}=\lambda \overrightarrow{AC}$.
Other properties if $A$, $B$ and $C$ are colinear:
$$\left| \frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{\left|\overrightarrow{AB}\right|\cdot\left|\overrightarrow{AC}\right|} \right| = 1$$:\
$$\overrightarrow{AB}\times\overrightarrow{AC} = \overrightarrow{0}$$
Also, two ways to write the equation of a line in 3D:
$$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$$
where $(x_0,y_0,z_0)$ is a point on the line and $(a,b,c)$ is the direction vector of the line, provided that $abc\ne 0$.
$$
\begin{align}
x&=x_0+at,\\
y&=y_0+bt,\\
z&=z_0+ct.
\end{align}$$
All that remains is calculation.