I doubt that Othello suffered from impostor syndrome. If he did, I would expect this to be reflected in at least some of his speeches.
When he speaks in his professional capacity as general in the Venetian military, his speeches are metrically very regular, which, combined with their actual content, can be interpreted as a mark of self-confidence. From the point of view of content, he know that the Venetian needs him because of the services he has rendered them. See for example, Act 1, scene 2:
Let him [Brabantio, Desdemona's father] do his spite:
My services which I have done the signiory
Shall out-tongue his complaints.
Commenting on Act 1, scene 3, Norman Sanders writes in his edition of the play (New Cambridge Shakespeare, 1984; emphasis added):
Every line Othello utters in his normal manner illustrates how all he has experienced has been related to himself with a breathtaking egocentricity which is clearly signalled by the grammatical focus of his avowal of the sincerity of his love for Desdemona: "She loved me for the dangers I had passed, / And I loved her that she did pity them" (1.3.166-7).
Such egocentricity hardly seems compatible with the self-doubt of someone with impostor syndrome. If Shakespeare had wished to make us think Othello experienced impostor syndrome, I would have expected soliloquies that counter the above impression, e.g. telling us that his public attitude is just bravado. However, it is Iago who gets almost all the soliloquies, which expose him as a Janus-faced schemer.
(In The Story of the Night: Studies in Shakespeare's Major Tragedies (1961), John Holloway showed that Othello's manner and speech conform with the sixteenth-century conduct manuals on how princes should behave.)
For evidence of an impostor syndrome in his private life, one would look at the speeches in which he talks to or about Desdemona, i.e. to look for indications that he might not feel he deserves her. For example, this is how he greets Desdemona after his arrival on Cyprus (Act 2, scene 1):
It gives me wonder great as my content
To see you here before me. O my soul's joy!
If after every tempest come such calms,
May the winds blow till they have waken'd death!
(...)
If it were now to die,
'Twere now to be most happy; for, I fear,
My soul hath her content so absolute
That not another comfort like to this
Succeeds in unknown fate.
I man who believes he is not worthy of his wife would not be so unreservedly relieved and happy at seeing her again; his relief would be dampened by his self-doubt.
A more interesting speech is his soliloquy in the middle of the play, after Iago has started sowing the seeds of doubt (Act 3, scene 3):
Haply, for I am black
And have not those soft parts of conversation
That chamberers have, or for I am declined
Into the vale of years,--yet that's not much--
She's gone. I am abused; and my relief
Must be to loathe her. O curse of marriage,
That we can call these delicate creatures ours,
And not their appetites! I had rather be a toad,
And live upon the vapour of a dungeon,
Than keep a corner in the thing I love
For others' uses. Yet, 'tis the plague of great ones;
Prerogatived are they less than the base;
(...)
The first few lines mention him being black and older than Desdemona, but he dismisses these reasons saying, "yet that's not much". Instead of feeling loathed, he says, "my relief / Must be to loathe her", i.e. the negative feelings are now aimed at Desdemona instead of at himself. Then they seem to shift back to himself: "That we can call these delicate creatures ours, / And not their appetites! I had rather be a toad". However, the reason he gives for this is Desdemona's unfaithfulness, not a belief in his own (real or imagined) shortcomings, since he continues, "'tis the plague of great ones"; these are not the words of someone who doubts his own accomplishments.
Finally, there are his last words before he stabs himself in Act 5, scene 2:
Soft you; a word or two before you go.
I have done the state some service, and they know't.
No more of that.
The tone is now of one who is used to giving commands but who know that he has been horribly deceived: "Soft you" commands silence rather than politely requesting it; "they know't" sounds like an allusion to his words on Act 1, scene 2 cited above. He then turns his attention to what happened in his private life:
I pray you, in your letters,
When you shall these unlucky deeds relate,
Speak of me as I am; nothing extenuate,
Nor set down aught in malice: then must you speak
Of one that loved not wisely but too well;
(...)
He asks those around him to describe himself and his actions as objectively as they can. He regains his dignity and nobility, not just by taking responsibility for his mistakes but by doing this in measured language using mostly end-stopped lines. I think it would be difficult for someone experiencing impostor syndrome to show so much dignity in their last words, just before they take their life.
