We have
$$\log \sum_{k \ge 0} T_k t^k = \sum_{i=1}^n \log \frac{x_i t}{1 - e^{-x_i t}}$$
so if we write
$$\log \frac{x_i t}{1 - e^{-x_i t}} = \log \sum_{k \ge 0} B_k^{+} x_i^k \frac{t^k}{k!} = \sum_{k \ge 1} b_k x_i^k \frac{t^k}{k!}$$
(using the sign conventions explained on Wikipedia) then we straightforwardly have
$$\sum_{k \ge 0} T_k t^k = \exp \left( \sum_{k \ge 1} b_k p_k \frac{t^k}{k!} \right).$$
WolframAlpha gives that the generating function of the series $b_k$ begins
$$b(t) = \log \frac{t}{1 - e^{-t}} = \frac{t}{2} - \frac{t^2}{24} + \frac{t^4}{2880} - \frac{t^6}{181440} \pm $$
which gives $b_2 = - \frac{1}{12}, b_4 = \frac{1}{120}, b_6 = - \frac{1}{252}$. Plugging the denominators into the OEIS gives A006953, the sequence of denominators of $\frac{B_{2k}}{2k}$, which suggests the following. We have
$$b'(t) = \frac{d}{dt} \left( \log t - \log (1 - e^{-t}) \right) = \frac{1}{t} - \frac{e^{-t}}{1 - e^{-t}} = \frac{1}{t} \left( 1 - \frac{t}{e^t - 1} \right) = - \sum_{k \ge 1} B_k^{-} \frac{t^{k-1}}{k!}$$
which gives $b_k = - \frac{B_k^{-}}{k}$ for $k \ge 1$. Altogether we have
$$\boxed{ \sum_{k \ge 0} T_k t^k = \exp \left( \sum_{k \ge 1} - \frac{B_k^{-}}{k} p_k \frac{t^k}{k!} \right) }.$$
As a sanity check, expanding the terms in WolframAlpha up to $t^4$ in terms of elementary symmetric polynomials / Chern classes gives
$$T_1 = \frac{c_1}{2}$$ $$T_2 = \frac{c_1^2 + c_2}{12}$$ $$T_3 = \frac{c_1 c_2}{24}$$ $$T_4 = \frac{-c_1^4 + 4c_2 c_1^2 + c_3 c_1 + 3c_2^2 - c_4}{720}$$
which agrees with the formulas for the first few terms of the Todd class on Wikipedia.
Edit: I believe this is the same as the result you quote except that the linear term in your result is zero (equivalently, $c_1$ vanishes for a hyperkahler manifold).