Using being in the middle of sentences [duplicate]

As for the pronoun, both him and his -- respectively, the ACC-ing complementizer and the POSS-ing complementizer, as they're called in the trade -- are acceptable as the subject of the gerund complement clause.

POSS-ing is slightly more formal and more often written, and may be claimed to be "more grammatical" or "the only correct choice" or something of the sort. But it's your choice, really.

As for the parse, there are 3 clauses, because there are three verbs: want, destroy, achieve

There is, as usual, one main clause (S₀)

• S₀ = He does not want S₁

with an A-Equi infinitive complement clause S₁as the direct object of want:

• S₁ = [for him] to destroy his parents' dream of S₂

and a gerund complement clause S₂ as the object of the preposition of:

• S₂ = his/him achieving a Cambridge degree.

The ACC-ing structure may be more appropriate than the POSS-ing structure on occasion, and vice versa.

We watched him leaving the building to see if he remembered to lock up.

We expected his leaving the company to take place long before it actually did.

The variant with him focuses more on the person, the variant with his more on the event. In the above two examples, this strongly suggests, or dictates in the second case, which variant should be used. In the original, I think him (more personal) just shades it. Either is grammatical.

.................

A member mentioned the fact that some educational establishments regarded the ACC-ing construction as ungrammatical in the fairly recent past. The only historical treatment I've found is from Nonfinite Structures in Theory and Change By D. Gary Miller, which includes

Around [1400], NOM-ing (replaced by ACC-ing in about 1900) split off from POSS-ing....

This implies that school grammars of the mid-20th Century which forbade the ACC-ing structure might be seen as hyperprescriptivist. Which comes as little surprise.

You’re on the right lines in thinking in terms of postmodification. The whole of of him achieving a Cambridge degree postmodifies the noun dream in that it answers the question ‘What kind of dream?’ The dream is about him doing something, the something being achieving a Cambridge degree.