I know that a dataset was generated $2/5$ of the times using a standard Gaussian and $3/5$ (which we call $C_1$) of the times using a student #t# with parameter $\nu = 1$ (we call this $C2$). I want to use Bayes' Theorem to compute the probability that the dataset was generated by this or that distribution. In order to do this, I would take the first element of my data set (though order doesn't matter), compute
$$ P(D=C_i|x)= \frac{P(x|D=C_i)P(D=C_i)}{P(x|D=C_1)P(D=C_1)+P(x|D=C_2)P(D=C_2)} $$
and then proceed in the same way on the next element, substituting to the $P(D=C_i)$ the posterior I just found, $P(D=C_i|x)$. Is this approach correct?
I assume that you want to find the probability that a particular observation came from the standard normal distribution rather than the $t$ distribution.
Let's call the density of the normal distribution $\phi(x)$, and the density of the $t$ distribution $f_t(x)$. We can use these as likelihoods in Bayes theorem and say $$\mathbb P(\text{Normal} \mid x) = \dfrac{\frac25 \phi(x)}{\frac25 \phi(x) +\frac35 f_t(x)}$$
I think this is what you have done, except that where you wrote $P(x\mid D=C_i)$ you have a minor typo in the subscript, and more importantly you should be using a density and (as Matthew Pilling commented) not a probability
If you are now saying the whole dataset was generated by just one of the distributions (with the $\frac25:\frac35$ prior as to which one) then you want $$\mathbb P(\text{Normal} \mid \mathbf x) = \dfrac{\frac25 \prod\limits_i \phi(x_i)}{\frac25 \prod\limits_i \phi(x_i) +\frac35 \prod\limits_i f_t(x_i)}.$$ You could do this iteratively but you get the same answer more quickly doing it all at once