We know that Julius Caesar was liked by the people early in the play when Flavius and Marullus chastise the common people for celebrating and for putting decorations on Caesar's statue. They ask, "Knew you not Pompey?" reminding the people that they had formerly loved Pompey, conquered by Caesar before the play begins. Marullus and Flavius tell the plebians to remove the decorations from Caesar's statues.
Much later, after the conspirators stab Caesar, Brutus momentarily persuades the populace that the assassination of Caesar was justified. He gets them thinking, if only for a moment, that maybe Caesar was too ambitious. But then Marc Antony delivers a more powerful oration, persuading the people that Caesar did, indeed, do much for them (leaving them parks, a monetary share of his will and more) and that they should exact revenge for his death. The mob is fickle, to be sure...one moment they are pleased with Caesar's death and the next moment they mourn him...but it appears that their predominant feelings were positive toward the ruler. After Antony's rousing funeral oration, she even kill Cinna, the poet because he has the same name as Cinna, the conspirator. Their love for Caesar may be fickle and changeable, but they follow Antony's lead, leaving the conspirators in mortal danger.
Julius Caesar the man, as opposed to the Shakespearean character, was liked by the people. He was a member of the populares, a group of the Roman political elite that pushed for reform and used populist policies to gain power through the mob. In modern American terms he would be regarded as a liberal. Thus, the very fact that Julius Caesar came to power was due in part to his support from t the common people. It was also the reason that he was killed, as even some of his supporters in the senate didn't like that he was levering his popular support in order to become emperor.
Dan Carlin did a superb podcast on the end of Republican Rome which will give you a good understanding of the period (they'll cost you a couple bucks but are well worth it) and I would also recommend Mike Duncan's History of Rome series.
Julius Caesar was liked by the people. He was offered three times for the crown. When he was killed, Antony told people about Caesar will left to the people of Rome. Before Antony gave speech, the people agreed with Brutus speech that Caesar was ambitious. Caesar left land and money to the people. When people found out, they were mad with the conspirator. The people changed their mind a lot about Caesar.