Let $R$ be a DVR with a fixed uniformizer $t$. My main considering example is the local ring of an algebraic curve over an algebraically closed field $k$, so let's assume $R$ is a $k$-algebra and has the residue field isomorphic to $k$.
ex-question: Is it true that the $t$-adic completion of $R$ is isomorphic to $k[[t]]$?
Update: The answer is yes, see https://mathoverflow.net/questions/191725/completion-of-a-local-ring-of-a-curve.
So it remains the Main question: How to explicitly write elements of $R$ as power series of $t$? In particular, how to write another uniformizer into a power series of $t$?
Let $r \in R \cong k[\![t]\!]$. Let $v: R \to \mathbb N \cup \{ \infty \}$ denote the ($t$-adic) valuation. Then we have $$ r = \sum_{i=v(r)}^\infty a_it^i, $$ and if I understand your question correctly, you are asking how to find the $a_i$. This can be done recursively as follows. Let $p: R \to k$ be the projection to the residue field. Then we have $a_{v(r)} = p(r/t^{v(r)}) \in k$, and for $n > v(r)$, $$ a_n = p\left((r - \sum_{i=v(r)}^{n-1}a_it^i)/t^n\right). $$