I suppose that Beilinson's compact generator (and, in fact, tilting object) $\mathcal{O} \oplus \mathcal{O}(1)$ in $D(\mathbb{P}^1)$ is the most well known example. I have the following simple construction for compact generator (but not a tilting object) on projective line. Lets take any point $z \in \mathbb{P}^1$ and denote skyscraper sheaf at this point $k(z)$ then $$ C=\mathcal{O} \oplus k(z) $$ is a compact generator. Is that correct?

It is correct. Taking the cone of a map $k(z) \to O[1]$ you get $O(1)$. Thus both $O$ and $O(1)$ are contained in the subcategory generated by $C$, and since these two generate $D(P^1)$, the same is true for $C$. 

