*To provide some context*, first let $\Omega$ be a bounded domain in $\mathbb{R}^3$ with smooth boundary.

The popular way of constructing weak solutions is by Galerkin approximation with the eigenfunctions of the Stokes operator (due to Hopf in 1951, or see Ladyzhenskaya's [book][1] Chapter 6, Section 6). A second way is to solve the mollified equations
$$ u_t + \text{div} (u \otimes u_\rho) - \Delta u + \nabla p = f, $$
where $u_\rho$ is the mollification of $u$ with an appropriate test function. This is what Leray did for the unbounded case, but it is possible in a bounded domain as well.

In either case, you obtain a sequence of functions $u^{(N)}$ which is bounded in $L^\infty_t L^2_x(\Omega \times (0,T))$ and $L^2_t H^1_{x,0}(\Omega \times (0,T))$. By boundedness, there is a subsequence $u^{(N)}$ converging in an appropriate sense (weak-$\ast$ in the former case, weak in the latter).

Others have remarked that the above modes of convergence are not enough to obtain a subsequence converging pointwise a.e. (indeed, for the target space $\mathbb{R}$, weak convergence in $L^2(0,T)$ not does imply pointwise convergence a.e.).

*To obtain pointwise a.e. convergence in time*, we can prove strong convergence in a space like $L^2_t L^2_x(\Omega \times (0,T))$. To do this, consider the [Aubin-Lions lemma][2] with $p=2$ and the triple
$$ H_0^1 \overset{\text{cpt}}\hookrightarrow L^2 \hookrightarrow H^{-3}.$$
The compactness of the first embedding follows from the boundedness of $\Omega$ and the Rellich-Kondrachov theorem. Aubin-Lions tells us that
$$ W = \{ f \in L^2_tH_0^1(\Omega \times (0,T)) : f_t \in L^2_tH_x^{-3}(\Omega \times (0,T)) \} \overset{\text{cpt}}\hookrightarrow L^2_tL^2_x(\Omega \times (0,T)).$$
I choose to bound the derivative in $H_x^{-3}(\Omega)$ because I want the use of the Sobolev embedding theorem in $n=3$. The nice thing about Aubin-Lions is that you can take as negative a Sobolev space as you want.

To meet the requirements of Aubin-Lions, we remark that the sequence $u^{(N)}$ is already bounded in $L^2_tH^1_{x,0}$. It remains to bound the derivative in $L^2_tH_x^{-3}$. Something like the following should be attainable from the equation:
$$ || \partial_t u^{(N)} ||_{L^2_t H_x^{-3}} \lesssim ||u^{(N)}||_{L^\infty_tL^2_x}^2 + ||u^{(N)}||_{L^2_t H^1_x} + ||f^{(N)}||_{L^2_t H^{-1}_x} \lesssim 1. $$
Aubin-Lions then tells us you can choose a subsequence such that
\begin{align}
u^{(N)} \to u &\text{ in } L^2_tL^2_x(\Omega \times (0,T)) \\
u^{(N)} \overset{\ast}\rightharpoonup u &\text{ in } L^\infty_tL^2_x(\Omega \times (0,T)) \\
u^{(N)} \rightharpoonup u &\text{ in } L^2_tH^1_x(\Omega \times (0,T)).
\end{align}
The strong convergence gives you pointwise a.e. convergence on the time interval, which is what you asked for.

Please note: I have essentially given you a poor rewording of the argument in [Seregin's lecture notes][3] (see Chapter 5), and I am mostly just rerecording it here. There is a preprint of his notes which comes up on Google, if you want.

For the case $\Omega = \mathbb{R}^3$, the Aubin-Lions lemma about compactness is not available. In such case, you can look at what Leray did in his original 1934 paper, which is available [here][4] in English and French.


  [1]: http://www.amazon.com/Mathematical-Theory-Viscous-Incompressible-Flow/dp/1614276714
  [2]: https://en.wikipedia.org/wiki/Aubin%E2%80%93Lions_lemma
  [3]: http://www.amazon.com/Lecture-Regularity-Theory-Navier-Stokes-Equations/dp/9814623407
  [4]: http://www.math.cornell.edu/~bterrell/leray.pdf