Suppose I have a map $f:X\rightarrow Y$ continuous, $X$ is compact, connected and also $f$ is a local homeomorphism, what condition should we include in $X$ so that $f$ becomes a covering map? Am I making any sense?
If $X$ and $Y$ are Hausdorff, then compactness of $X$ and surjectivity of $f: X\to Y$ are already sufficient conditions for $f$ to be a covering map (edit: just to be clear, this is in addition to $f$ being a local homeomorphism). See this question.
Note that $f$ is necessarily surjective if $Y$ is assumed to be connected, because $f$ is closed and open (still under the hypothesis that $Y$ is Hausdorff).