1. **Define the points $p$ and $q$:**
- Let $p = (\cos \theta, \sin \theta)$ where $\theta$ is uniformly distributed over $[0, 2\pi)$.
- Let $q = (r \cos \phi, r \sin \phi)$ where $r$ is uniformly distributed over $[0, 1]$ and $\phi$ is uniformly distributed over $[0, 2\pi)$.

2. **Determine the coordinates of the vertices of the rectangle $R$:**
- The rectangle $R$ has sides parallel to the $x$- and $y$-axes with diagonal $pq$.
- The vertices of $R$ are:
$$(\cos \theta, \sin \theta), (r \cos \phi, \sin \theta), (\cos \theta, r \sin \phi), (r \cos \phi, r \sin \phi)$$

3. **Condition for $R$ to lie inside $C$:**
- For $R$ to lie entirely within the unit circle, all vertices must satisfy the equation $x^2 + y^2 \leq 1$.
- The vertices are:
$$(\cos \theta, \sin \theta), (r \cos \phi, \sin \theta), (\cos \theta, r \sin \phi), (r \cos \phi, r \sin \phi)$$
- We need to check the conditions:
$$(r \cos \phi)^2 + (\sin \theta)^2 \leq 1$$
$$(\cos \theta)^2 + (r \sin \phi)^2 \leq 1$$
$$(r \cos \phi)^2 + (r \sin \phi)^2 \leq 1$$

4. Simplify the conditions:
- The third condition simplifies to:
$$r^2 (\cos^2 \phi + \sin^2 \phi) \leq 1 \implies r^2 \leq 1 \implies r \leq 1$$
This condition is always satisfied since $r$ is uniformly distributed over $[0, 1]$.

5. Evaluate the remaining conditions:
- The first condition:
$$r^2 \cos^2 \phi + \sin^2 \theta \leq 1$$
Since $\sin^2 \theta \leq 1$, this condition is always satisfied for $r \leq 1$.

- The second condition:
$$\cos^2 \theta + r^2 \sin^2 \phi \leq 1$$
Since $\cos^2 \theta \leq 1$, this condition is always satisfied for $r \leq 1$.

6. Conclusion:
- Since all conditions are satisfied for $r \leq 1$, the probability that no point of $R$ lies outside of $C$ is 1.

The final answer is $\boxed{1}$.