In Shakespeare's play, Antony has many reasons for approaching and befriending the conspirators after they have assassinated Caesar. The first, and most practical, is to be sure that his (Antony's) life is save and that the conspirators are not after him as well. He also wants to know why the conspirator committed such a deed. He shakes each of their bloody hands to mark the conspirators and guilty and remember who was involved. Lastly, he gets permission from Brutus to give Caesar a proper burial and to speak at Caesar's funeral.
While it may seem like Antony is becoming subservient to the conspirator and Brutus, Antony is already two steps ahead of them, and planning their demise (or at least Shakespeare was). Antony wants to "keep his friends close and his enemies closer". He wants to be able to speak at Caesar's funeral and be able to display the body to provoke the plebians. He agrees to Brutus' terms--that Antony speak no ill will on the conspirators--but Antony is a shrewd public speaker and will twist the words to do his will.
A reader can tell that Antony planned all of this by the speech Antony makes to Caesar's body once the conspirators have left. He asks Caesar to forgive him for being meek and friendly with these butchers and promises horrid revenge on them.
Actually, Antony only appears to befriend the conspirators after Caesar's assassination. Antony was close to Caesar, both politically and in a social sense.
Prior to Caesar's assassination, at a meeting of the conspirators at Brutus' house, Cassius suggests that, in addition to killing Caesar, the conspirators take out Antony as well, thereby eliminating a potential future political threat. Brutus refuses to take part is killing Antony (a poor decision, as it turns out later).
After Caesar's assassination, Antony sends a messenger to the conspirators for safe passage to the [scene of the crime]. Upon his arrival, he is surrounded by the conspirators that murdered his beloved Caesar. Appearing to join them was most likely for survival purposes.
He wants to avenge Caesar's death, but he is outnumbered.
Although Cassius expresses outrage at Antony's request to speak at Caesar's funeral, Brutus allows it (another poor decision), but Antony later appeals to the crowd's emotions by listing all Caesar--whom the conspirators claimed was "ambitious"--had left to the people. This enraged the crowd to the point that it wanted vengeance, blood, the death of the conspirators.