Show that kleene plus language is a regular language by describing how you build a DFA or NFA for it
You can use Thompson's construction. It constructs NFA with $\epsilon$ moves. But remove the forward $\epsilon$-transition from the new start state into the new final state since you are interested in $L^+$ and not in $L^*$.

NFA with $\epsilon$ moves is a generalization of NFA and DFA. But you can systematically transform any NFA with $\epsilon$ moves into equivalent NFA without $\epsilon$ moves and DFA. Look here for more.