Confusing example of a language which may be Context-free or not context-free
The best way to show that a language is context-free is to use one of the equivalent definitions of context-free languages. The two most popular ones are:


A language is context-free if it is generated by a context-free grammar.
A language is context-free if it is accepted by a (non-deterministic) push-down automaton.


The language $\{ a^n b^n : n \geq 0 \}$ is generated by the grammar $S \to aSb | \epsilon$. Let's try to use the same idea for your language: $S \to 0^k S 1 | \epsilon$ (we can implement $0^k$ using context-free rules). The problem is that we can't guarantee that the same $k$ is used in all productions. This leads me to conjecture that the language is not context-free.

How do we show that a language is not context-free? The most popular way is to use the pumping lemma or one of its variants, though this method doesn't always work. Perhaps you should give it a try.