In words: to show a portfolio is self-financing, you want to show that the (instantaneous) change in the value of the portfolio is equal to the change in the value of your cash/bank account and risky asset(s), i.e.:
$$ dV_t = \phi_t dS_t + \psi_t dB_t$$
where \( \phi_t \) is the quantity of the risky asset, and \( \psi_t \) is the quantity of cash/bank account. (NOTE: this is bookwork – cf. Alex's slides). This is exactly what you want to do in the tutorial question.
For the tutorial question in particular, you can either start with \( dV_t \) and show that this equals \( \phi_t dS_t + \psi_t dB_t \), or you can do it in the opposite order (i.e. start with \( \phi_t dS_t + \psi_t dB_t \) and go from there).
Furthermore, you need to use some of the additional information given in the question.
From there, it's mostly a matter of manipulation of the expressions (e.g. using Ito's lemma) and substitution to show that self-financing holds. As such,
i) If you start with \( dV_t \), then the key thing to note is that
$$ B_t E_t = e^{rt}e^{-rt}V_t = V_t $$
and so
$$ dV_t = d(B_t E_t) $$
ii) if you start with \( \phi_t dS_t + \psi_t dB_t \), then the key thing to note is that
$$B_t D_t = e^{rt} e^{-rt} S_t = S_t $$
and so
$$dS_t = d(B_t D_t) $$
Remember that all of the definitions of the terms in the above expressions are given in the question – use them as such.