World number one Williams faces fellow American Vania King in second round action and a victory would put her five match wins away from breaking the professional era record for grand slam titles that she currently shares with Steffi Graf.
"I can improve a lot," said Williams, who needed just 63 minutes to get by Russia's Ekaterina Makarova in the first round. "I can get a lot better. I feel like there's much, much more I can do. That's the only thing I can do is do that."
Williams came two wins away from a rare calendar-year grand slam last year only to lose in the U.S. Open semi-finals to eventual runner-up Roberta Vinci of Italy.
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This year, with a niggling shoulder injury knocking some speed off her usually destructive serve, Williams is not sure what she is capable of.
"I had a great experience last year,” she said. “I was going for something that no one has done in a really long time. Yeah, it didn't end out wonderful for me or the way I wanted it to end. But it was all I could do. That's all I could do.
"If I could make the semis this year, I'd be excited about that. I need to at least do something."
Fourth seed Agnieszka Radwanska, one of three women who could unseat Williams atop the world rankings following the U.S. Open, meets Britain's Naomi Broady.
Fifth seed Simona Halep faces former French Open finalist Lucie Safarova while sixth seed Venus Williams meets Julia Goerges of Germany.
On the men's side, Wimbledon and Olympic champion Murray plays Marcel Granollers of Spain. Murray has won six of seven career meetings with Granollers and should enjoy smooth passage to the third round.
In other action on the men's side, two-time grand slam winner Stan Wawrinka faces Italy's Alessandro Giannessi while Japanese sixth seed Kei Nishikori plays Russia's Karen Khachanov.
(Editing by Frank Pingue)
