It is actually kind of hard to say what Julius Caesar’s supporters would have felt about Marc Antony’s involvement with Cleopatra. For one thing, the surviving sources we have are heavily biased against Cleopatra, who is portrayed as a foreign threat to Roman power in the Mediterranean and a promiscuous schemer. Most primary literary sources for Cleopatra come from the Augustan era, and are heavily influenced by Augustus’ politics and propaganda.
#What Augustan sources generally say
Many leading scholars, such as Duane Roller, point to undercurrents of pro-Cleopatra sentiments in Roman literature as evidence that she actually was not as maligned as we might think. Instead, we are only seeing the sanitised, “official” opinions of the day, not necessarily an accurate representation of popular sentiments. Nevertheless, they are our only insight into contemporary reactions to Antony and Cleopatra.
Generally, Antony is painted as weak-willed at best, traitorous at worst for his relationship with Cleopatra and his alleged submission to her political agenda. But the very condemnation of Antony on account of his love for Cleopatra creates a certain awkwardness in Augustan literature, since Caesar loved her first. The inappropriateness of the married Julius Caesar carrying on with the (also technically married) Cleopatra is treated with disapproval in Roman sources, but it pales in comparison to Antony’s polygamous marriage to Cleopatra while still married to Octavia. The aversion to polygamy and the hostile reaction to Antony’s political concessions to Cleopatra at events like the Donations of Alexandria escalate the issue from a matter of morality and cultural values, to an actual war. By allowing Cleopatra to threaten the very security of Rome, Antony fails the (generally more favourable) legacy of Caesar as a politician, while Augustus lives up to and surpasses Caesar’s legacy.
In Roman poetry and biographies from the late 1st Century BCE onwards, the love of Cleopatra is portrayed as a folly and a vice shared by both Caesar and Antony. For example, epics like Lucan’s Pharsalia foreshadow the downfall of Antony with Caesar’s own failure to resist Cleopatra’s advances in 47 BCE. The difference is that Caesar’s passions are not blamed for his downfall, only for major errors in judgement. Antony on the other hand is portrayed as being utterly destroyed by the mistakes he made on Cleopatra’s account. So the two are similarly portrayed as willing victims of a malicious seducer, but Antony is faulted for allowing it to go much farther.
But is this actually how their contemporaries saw it, or how Augustus wanted it to be seen?
#The other half of it
For one thing, Cleopatra did have some popularity in Rome, at least before all the rivalry with Octavian. Some statesmen like Cicero were quite hostile towards her, but there were also a number of prominent Romans, particularly but not only followers of Antony, who were on friendly terms with Cleopatra. Only a few of these are identified by name in Roman sources, and only one is named (in a corrupted and damaged) Egyptian papyrus. Even after her death there were evidently those who took a more nuanced view to her life and legacy.
Secondly, those who did see her as a dangerous influence were apparently concerned about Caesar’s politics with regards to her. Caesar certainly was not without critics, some of whom actually stabbed him to death out of concerns that he was a tyrant, so maybe Antony's flawed legacy has some comparisons. That he was criticised by his contemporaries for his involvement in the Alexandrian Civil War on her behalf is hinted at by contemporary sources, and stated out right by later sources like Plutarch.
There were even rumours that Caesar had married her or at least intended to, and these combined with other rumours about his monarchical ambitions. Nicolaus of Damascus reports a rumour that Caesar planned to establish his own senate in Alexandria and make Caesarion his heir, rumours rather similar to those that later circulated about Antony. Of course these rumours are treated as false, but the very fact that a client of Augustus included these in his biography means that they must have been circulated by somebody.
#The personal aspect of Caesar’s memory
There were not really any suggestions that Antony was being disrespectful or disloyal to Caesar’s memory by entering into a relationship with Caesar’s former lover, especially since Caesar was long dead. Part of the reason for this might be Roman (and Greek) attitudes towards sex and marriage in general. For one thing, since Cleopatra and Caesar were never legally married, the same stigma of adultery would not have attached itself to any other relationships they had. Which is to say that, the illicit nature of Caesar and Cleopatra’s relationship also precluded them from having a legal obligation of fidelity, and to some extent also precluded them from a moral obligation.
Perhaps more importantly, in all likelihood several years elapsed between Caesar’s death and any kind of relationship between Antony and Cleopatra. So regardless of how Caesar’s relationship with Cleopatra was perceived, it was (ahem) ancient history. The high mortality rates and typically young age of marriage in Greco-Roman society meant that serial monogamy was the norm, not the exception, and there was nothing wrong with marrying a man’s widow, let alone his ex-mistress.
#Antony as step-father to Caesar’s son
Antony’s behaviour towards Caesarion, the illegitimate son of Cleopatra, is the other important aspect of your question since Caesarion was supposedly fathered by Caesar. Antony took on the role of Caesarion’s step-father with enthusiasm, no doubt due to both his relationship with the latter's mother as well as the connection it gave him to Caesar’s legacy and image. However, Caesarion’s paternity was not universally agreed upon, and Caesar’s own avoidance of the issue meant that his friends actually argued the point later on. Even today, there is some controversy over Caesarion’s conception, although it usually assumed that he was Caesar’s son. Jerome Carcopino even suggested that Antony was the biological father of Caesarion, but this is generally discounted.
To those who agreed that both Caesar was Caesarion’s father and that Caesarion’s paternity mattered at all, Antony’s behaviour towards Caesar’s biological son would only have been right. But this was almost certainly not a popular sentiment. Even without Octavian and his supporters claiming that Caesarion was another's child, there was never a lot of love in Rome for a bastard born to a foreign woman who probably was not a Roman citizen (yet, anyway). If anything, Caesar having a child with a foreign queen was embarassing, especially while he failed to have a legitimate child with his wife.
Moreover, for all the rumours and claims thrown around by Cleopatra’s supporters and Caesar’s detractors, Julius Caesar never formally acknowledged Caesarion or designated him as his heir. First Antony was designated as his successor, and this later changed to his grand-nephew Octavian (posthumously adopted as his son). Because of this, Antony’s promotion of Caesarion as Caesar’s true son was seen as a slap in the face to Caesar’s adopted son Octavian, and an insult to Caesar’s legacy.
#Antony and Caesar’s legacy overall
In reality, both Caesar and Antony had their own reasons to enter a relationship with Cleopatra. Not only was she perhaps the most illustrious monarch in the 1st Century BCE (able to claim both Ptolemaic and Seleucid heritage), she was also fabulously wealthy and in charge of one of the largest independent kingdoms friendly to Rome. As far as allies went, Cleopatra was a valuable friend to both men if they wanted to promote their own interests in the eastern Mediterranean. Not to mention the fact that she was a tangible connection to the legacy of Alexander the Great.
Antony certainly had Caesar’s plans to conquer the Parthian Empire in mind when he enlisted Egypt’s aid in his own doomed Parthian campaign. Cleopatra, for her part, seems to have been somewhat reluctant to enter into an alliance with Antony and Octavian following Caesar’s assassination, preferring them only to Caesar’s assassins. Unfortunately it is rather hard to unravel Cleopatra’s activities after the death of Caesar and before her alliance with Antony.
Antony’s relationship to Caesar’s legacy in general is actually a bit complicated. I think it could be argued that his relationship to Caesar’s legacy was worse before Caesar died, and he became a more central figure after he became one of Caesar’s most powerful followers. On the one hand, Antony was Caesar’s relative and one who gained a good deal of prestige in war, but the two were not always on the best of terms. Antony aggressively promoted himself as Caesar’s successor and this brought him into conflict with Octavian. While Cleopatra’s promotion of Caesarion as Caesar’s heir was a factor in her rivalry with Octavian, her alliance with Antony was what tipped it over into a heated conflict.
At the end of the day, it becomes very difficult to separate the political propaganda of Antony and Octavian from the meat of our surviving evidence. There might not even be any point to trying since Caesar’s legacy was what those who came after made it. Hopefully this helps to shine a light on some facets of the issue.
#Sources
Cleopatra: A biography by Duane Roller
Cleopatra and Rome by Diane E. E. Kleiner
Antony & Cleopatra by Adrian Goldsworthy
Life of Augustus by Nicolaus of Damascus
Roman History by Cassius Dio
Life of Antony & Life of Caesar by Plutarch
Life of Caesar by Suetonius