Without being able to actually see Caesar and check for ourselves, we must rely on our literary and sculptural depictions of the man, but it seems safe to say that Caesar was in excellent shape. Caesar cultivated his appearance, for which he was accused of effeminacy. Suetonius describes his physical fitness thus:

Fuisse traditur excelsa statura, colore candido, teretibus membris, ore paulo pleniore, nigris vegetisque oculis, valitudine prospera, nisi quod tempore extremo repente animo linqui atque etiam per somnum exterreri solebat

It is said he was tall in stature, with a pale complexion, slender limbs, a face a little plumper, with bright black eyes, and good health, except when at the end of his life he used to suffer from fainting and had nightmares.

When Suetonius says that Caesar was "ore paulo pleniore" ("with a face a little plumper," than his limbs) he certainly doesn't mean fat--plenus doesn't mean that, it means full, and when used of a face it usually means something like "round." There is a shortage of portrait busts from Caesar's own lifetime, but all likenesses of him, contemporary or later, show him to have a perfectly ordinary face, even a little bony.

Plutarch has a similar description:

τὸ μὲν οὖν φιλοκίνδυνον οὐκ ἐθαύμαζον αὐτοῦ διά τὴν φιλοτιμίαν ἡ δὲ τῶν πόνων ὑπομονὴ παρὰ τὴν τοῦ σώματος δύναμιν ἐγκαρτερεῖν δοκοῦντος ἐξέπληττεν, ὅτι καὶ τὴν ἕξιν ὢν ἰσχνός καὶ τὴν σάρκα λευκὸς καὶ ἁπαλός καὶ τὴν κεφαλὴν νοσώδης καὶ τοῖς ἐπὶ τοῖς ἐπιληπτικοῖς ἔνοχος, ἐν Κορδύβῃ πρῶτον αὐτῷ τοῦ πάθους, ὡς λέγεται, τούτου προσπεσόντος, οὐ μαλακίας ἐποιήσατο τὴν ἀρρωστίαν πρόφασιν, ἀλλὰ θεραπείαν τῆς ἀρρωστίας τὴν στρατείαν, ταῖς ἀτρύτοις ὁδοιπορίαις καὶ ταῖς εὐτελέσι διαίταις καὶ τῷ θυραυλεῖν ἐνδελεχῶς καὶ ταλαιπωρεῖν ἀπομαχόμενος τῷ πάθει καὶ τὸ σῶμα τηρῶν δυσάλωτον.

And men did not marvel at his love of danger, on account of his ambition [more literally, "love of honors"], but his endurance of toils beyond the apparent ability of his body amazed them, since he lived a spartan life and had pale and tender skin and suffered from sickness of the head [epilepsy, which Caesar probably did not have] and was bound by epileptic fits, with the disease first falling on him, as they say, at Corduba. He did not make his weak health a pretext for softness, but made military campaigning the cure for his weakness, fighting off the disease and strengthening his body to be unconquerable with forced marches, simple living, constant camping, and enduring of toils.

Plutarch goes on to mention that Caesar barely slept, sleeping in his carriage so as not to waste time that could be spent working--this idea of Caesar as the tireless worker is ubiquitous in descriptions of Caesar's person, it's more or less his defining trait. Plutarch also mentions that Caesar was an accomplished horseman and marched at the head of the army--another constant feature of descriptions of the man is his manner of leadership by example.

Cicero, in one of his letters, also mentions of Caesar "ἐμετικὴν αγεβατ," ("of course he was on emetics"). This letter was written at the end of Caesar's life, when his health was supposedly failing. Moreover, it seems to be a compliment, and is probably supposed to mean that Caesar was, as Cicero might naturally expect from someone who knew Caesar, keeping a close eye on his physical health--emetics were frequently prescribed by the Hippocratics and their descended schools to maintain the proper balance of the humors

As for the impression of a fat, old Caesar, I can only imagine it derives from the general pop culture view of the Roman aristocrat as the decadent, orgiastic reveller that he was in reality most certainly anything but--Caesar, as the only Roman many people know anything about, would naturally be the prime target for such false abuse. As for where that impression comes from, that's an entirely different matter

I believe Mel Brooks portrayed him as heavier for comedic effect rather than any historical basis. He was actually in excellent health apart from migraines and epilepsy. He was personally concerned with his early hair loss but there is no reason to believe it was a byproduct of any other health concerns.

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