You can do it with a straightedge alone, though the lines tend to clutter the scene.
Start with the two solid black lines, with directions at your free disposal as long as you get four intersection points with the circle. (It helps to keep one line closer to the center of the circle and the other farther away.)
Then draw the dashed lines, then the blue line $p$. That blue line is called the polar of the point $P$. Interestingly, it does not depend on the particulars of the black lines you have begun with.
Now, if $P$ is outside the circle, then its polar $p$ crosses the circle, and the points of intersection are the points of tangency for tangents through $P$.
Bonus: This approach even works for a conic instead of a circle, as long as you are given at least five points of that conic. Takes even more lines though, unless the conic is drawn already, in which case it works the same way as for the circle. I have hinted at that elsewhere.