What was the physical appearance of Julius Caesar - historically and in Shakespeare's play?

Good question! The Roman historian Suetonius writes of Caesar that he

...is said to have been tall of stature with a fair complexion, shapely limbs, a somewhat full face, and keen black eyes; [...] He was somewhat overnice in the care of his person, being not only carefully trimmed and shaved, but even having superfluous hair plucked out, as some have charged; while his baldness was a disfigurement would troubled him greatly, since he found that it was often the subject of the gibes of his detractors...

They say, too, that he was remarkable in his dress; that he wore a senator's tunic with fringed sleeves reaching to the wrist, and always had a girdle over it, though rather a loose one; and this, they say, was the occasion of Sulla's mot, when he often warned the nobles to keep an eye on the ill-girt boy.

There are also (search in Google Images) lots of busts and statues which give us an idea of what the real Caesar looked like.

Shakespeare's Caesar is an old man, deaf in one ear and with some eyesight problems. He also "has the falling sickness" - suffers from epilepsy: a trait that some historians think he might have in common with the historical period. But Shakespeare, as ever, doesn't tell us a huge amount about his appearance!