I can discuss Shakespeare, but hopefully someone more knowledgeable about the historicity of Plutarch will chime in.

Julius Caesar is historically accurate in the sense that Shakespeare was (somewhat) faithfully following the narrative as written by Plutarch. In 1597 Sir Thomas North translated The Lives of Julius Caesar and Marcus Brutus into English. It was instantly popular, sold many copies in the London book shops, and was almost certainly used by Shakespeare as the basis for his play. Shakespeare wrote Julius Caesar some time in 1598 with its earliest known performance happening during The Globe Theatre's inaugural season on September 21st, 1599.

There are a number of notable similarities between Shakespeare's verse and Plutarch's narration. I will now quote part of an answer I gave to a previous question:

Shakespeare, in many cases, changes the narration but retains the essence of the story:

There Brutus, being afraid to be besieged, sent back again the noblemen that came thither with him, thinking it no reason that they, which were no partakers of the murder, should be partakers of the danger.

(Plutarch 126)

Cassius: And leave us Publius; lest that the people,

Rushing on us, should do your age some mischief.

Brutus: Do so; and let no man abide this deed

But we the doers.

(Julius Caesar 3.1.89-92)

Here we hear a difference in language, but see a continuity in story. Another example would be the battle scene where Brutus contemplates suicide:

Brutus as he sat bowed towards Clitus one of his men and told him somewhat in his ear; the other answered him not, but fell a-weeping. Thereupon he proved Dardanus, and said somewhat also to him. At length he came to Volumnius himself, and , speaking to him in Greek, prayed him…that he would help him to put his hand to his sword, to thrust it in him to kill him.

(Plutarch 170)

Brutus: Sit thee down, Clitus; slaying is the word,

It is a deed in fashion. Hark thee, Clitus.

Clitus: What, I, my lord? No, not for all the world.

Brutus: Peace then, no words.

Clitus: I’ll rather kill myself.

Whispering

Brutus: Hark thee, Dardanius.

Dardanius: Shall I do such a deed?

(Julius Caesar 5.5.4-8)

When Caesar is speaking to his wife about the nature of the omens she has observed, Shakespeare transmutes Plutarch’s narration into Caesar’s words:

And when some of his friends did counsel him to have a guard for the safety of his person, and some also did offer themselves to serve him, he would never consent to it, but said, it was better to die once than always to be afraid of death.

(Plutarch 78)

Caesar: Cowards die many times before their deaths;

The valiant never taste of death but once,

Of all the wonders that I yet have heard,

It seems to me most strange that men should fear;

Seeing that death, a necessary end,

Will come when it will come.

(Julius Caesar 2.2.32-37)

There are, of course, exceptions but almost the entire plot is lifted directly from Plutarch's writings into Shakespeare's play. Sometimes he takes the words straight out of Plutarch's mouth in a way that we today might consider plagiarism such as in the fifth act when Brutus' army is preparing to do battle:

…the greatest and chiefest things among men are most uncertain, and that, if the battle fall out otherwise today then we wish or look for, we shall hardly meet again, what are thou then determined to do…

(Plutarch 154)

Cassius: Now, most noble Brutus,

The gods to-day stand friendly, that we may,

Lovers in peace, lead on our days to age!

But since the affairs of men rest still incertain

Let’s reason with the worst that may befall.

If we do lose this battle, then is this

The very last time we shall speak together:

What are you then determined to do?

(Julius Caesar 5.1.93-100)

Another example:

…he should wear his diadem in all other places both by sea and land…

(Plutarch 90)

Casca: And he shall wear his crown by sea and land

(Julius Caesar 1.3.87)

There is also the matter of Plutarch's omens:

For, touching the fires in the element and…also the solitary birds to be seen at noondays sitting in the great market-place…men were seen going up and down in fire; and, furthermore, that there was a slave of the soldiers that did cast a marvelous burning flame out of his hand, insomuch as they that saw it thought he had been burnt, but, when the fire was out, it was found he had no hurt. Caesar self also, doing sacrifice unto the gods, found that one of the beasts which was sacrificed had no heart; and that was a strange thing in nature – how a beast could live without a heart.

(Plutarch 86,87)

Casca: But never till to-night, never till now,

Did I go through a tempest dropping fire…

And yesterday the bird of night did sit

even at noon-day upon the market-place,

Hooting and shrieking.

(Julius Caesar 1.3.9-13)

…a hundred ghastly women,

Transformed with their fear; who swore they saw

Men all in fire walk up and down the streets.

(Julius Caesar 1.3.23-25)

A common slave – you know him well by sight –

Held up his left hand, which did flame and burn

Like twenty torches join’d, and yet his hand,

Not sensible of fire, remaine’d unscorch’d.

(Julius Caesar 1.3.15-18)

Servant: They would not have you to stir forth today.

Plucking the entrails of an offering forth,

They could not find a heart within the beast.

(Julius Caesar 2.2.38-40)

I'm not going to type out all of these, but you get the idea: Shakespeare's plot and text rely heavily on his source material.

The two are not without their differences. Shakespeare compresses the events of several years into 5 days. He makes the Capitol the venue of Caesar's death rather than the Theatrum Pompeium. Shakespeare makes the Triumvirs meet in Rome instead of near Bononia. The period between Caesar's victory and Antony/Octavius regaining control is a span of 3 years. Caesar's assassination and the dueling speeches of Antony and Brutus happened months apart according to Plutarch while Shakespeare places all three events on one gigantically important day: The Ides of March. Shakespeare's final battle happens on the fifth day of the play, where Plutarch sets it months after the assassination.

All this compression was done in order to keep the story clipping along, which Shakespeare does with uncanny skill. Unlike Shakespeare, Plutarch is not known as a master of suspense. Shakespeare's real gift to the story is giving living, breathing, 3-dementional personalities to Cassius and Brutus that are not present in Plutarch. The Bard gives lip service to the sanctity of the crown in the character of Caesar, but in the end it is Brutus that best fits the description of the Shakespearean hero.

Sources:

Arden Shakespeare version of Julius Caesar

Lives of Julius Caesar and Marcus Brutus by Plutarch (Thomas North translation)

The Concise Oxford Companion to the Theatre by Phyllis Hartnoll

There was a post that was about the historical accuracy of Shakespeare's plays (wow has it been that long?) 9 months ago. The top post explains how Shakespeare more or less used Plutarch's record of the events but tweaked them to make the story a bit more fluid.

I hope this helps.