I`m trying to show that $\{\diamond\}$ is complete.
\begin{array}{ccc|c} P&Q&R&\diamond(P,Q,R)\\ \hline T&T&T&F\\ T&T&F&F\\ T&F&T&F\\ T&F&F&T\\ F&T&T&T\\ F&T&F&T\\ F&F&T&T\\ F&F&F&T\\ \end{array}
I know that $\{\vee, \neg \}$ is complete.
Then, if we show NOT & OR by $\diamond$ we done.
NOT is easy: $\neg p \Longleftrightarrow \diamond(p,p,p)$
However,is it correct to use T\F to show $p \vee q$, since I don`t know how is it possible to show it without T\F.
Thats is what I wrote: $p\vee q \Longleftrightarrow \neg(\diamond(T,p,q))$ is this correct?
What the right way to think about it when I need to show if is complete? It took my a lot of time to get to it and I not succeed to write it without using T or F.
Whether it's permissible to use $T$ and $F$ depends on your definition of "complete". See the discussion on the Wikipedia page for Functional completeness. But in this case, we don't have to use them: The term $\diamond(\lnot p,p,p)$ is always $T$.
Putting this together with your observations, we can define $p\lor q$ by $$\lnot(\diamond(\diamond(\lnot p, p,p),p,q)).$$