The “phishing” emails, disguised as speeding tickets, would have enabled the hackers to control her computer.
The infected computer would have sent information to at least three computers overseas, including one in Russia.
A spokesman for Mrs Clinton said there was no evidence of a breach.
The hacking attempts were included in thousands of emails released by the State Department.
Mrs Clinton’s opponents have accused her of putting US security at risk by using an unsecured computer system.
The five emails, sent over a four-hour period in August 2011, show hackers had Mrs Clinton’s email address, which was not public, and contained a virus concealed as a speeding ticket from New York state, where she lives.
The email containing instructions to open and print the speeding ticket misspelled the name of the city concerned, Chatham, came from a supposed New York City government account and contained a “Ticket.zip” file of the kind usually picked up by commercial antivirus software.
Nick Merrill, a spokesman for Mrs Clinton’s presidential campaign, said there was no evidence to suggest she replied to the emails or opened the attachment.
“All these emails show is that, like millions of other Americans, she received spam,” he said.
The presidential hopeful has admitted that her decision to use a private email server at her New York home was a mistake.
However, the latest set of her emails to be released also reveal frustration within the State Department at the technology it was using while she was in office.
In one email exchange Mrs Clinton’s then head of policy Anne-Marie Slaughter wrote that the department’s technology was “so antiquated” that high-level officials “routinely end up using their home email accounts to be able to get their work done quickly and effectively”.
She suggested writing an opinion piece to highlight the problem and Mrs Clinton agreed the idea “made good sense”, but her chief of staff Cheryl Mills warned against “telegraphing” how often senior officials relied on their private email accounts to do government business because it could encourage hackers.
Why did Hillary Clinton use her own email?
Mrs Clinton says the primary reason she set up her own email account was for “convenience” but sceptics say the real reason she did it was because it gave her total control over her correspondence.
How many emails?
According to Mrs Clinton, she sent or received 62,320 emails during her time as secretary of state – she says half of them were official and have been turned over to the state department.
Was it illegal?
Probably not. Mrs Clinton’s email system existed in a grey area of the law – and one that has been changed several times since she left office.
Why the controversy?
It’s a big deal because Mrs Clinton is asking the US public to trust that she is complying with both the “letter and the spirit of the rules”. Critics on the left and the right are concerned that she made her communications on sensitive nationalsecurity issues more susceptible to hackers and foreign intelligence services.