Is it possible to prove Tarski's Undefinability theorem from Turing's Halting Problem?
As cody observed, the problem with that strategy is that definable relations need not be computable. For example, the Halting Problem is definable in $(\mathbb{N}; +,\times)$ (see Kleene's $T$-predicate).

That said, a relativized form of the incomputability of the Halting Problem, applied to iterates of the Turing jump, does do the job:


First, we show that for each $n$ and each $\Sigma_n$ formula $\varphi$ the set $\{x: \varphi(x)\}$ is computable from ${\bf 0^{(n)}}$, uniformly in $n$ and $\varphi$.
Next, we show that for each $n$ the set ${\bf 0^{(n)}}$ is itself definable by some $\Sigma_n$ formula $\psi_n$. This is an application of Kleene's $T$-predicate.
Now we're good to go. Suppose we had a $\Sigma_n$ formula $\varphi$ defining the set of (Godel numbers of) true sentences in the language of arithmetic. Then ${\bf 0^{(n)}}$ could compute ${\bf 0^{(n+1)}}$ as follows: given $k$, we ask ${\bf 0^{(n)}}$ whether $\varphi(\underline{\ulcorner\psi_{n+1}(\underline{k})\urcorner})$ holds. (Here "$\underline{i}$" denotes the numeral corresponding to the number $i$, and "$\ulcorner\theta\urcorner$" is the Godel number of $\theta$.)
But that gives ${\bf 0^{(n+1)}}=({\bf 0^{(n)}})'\le_T{\bf 0^{(n)}}$, a contradiction by the relativized unsolvability of the Halting Problem.