Henry Richard Hammond, 27, remained in custody on Tuesday as his Sydney-based relatives issued a brief statement.

"Our family is in deep shock over the tragic death of Courtney Herron," the statement read.

"Our deepest sympathy and compassion goes out to Courtney's family and friends on their heartbreaking loss."

Hammond had experienced struggles for many years, the family added.

During the homeless man's court appearance on Monday, lawyer Bernie Balmer said his client had a history of mental health issues.

"There's a diagnosis of possible delusional disorder, possible autism spectrum disorder and historical diagnosis of ADHD," Mr Balmer said.

Hammond's family said he "grew up loved by his family and friends".

"Our hearts go out to Courtney's family and friends."

Ms Herron's tearful family have visited Royal Park in the inner-Melbourne suburb of Parkville, where her body was found by dog walkers on Saturday morning.

They left tributes for the 25-year-old who had been living on the streets and battling mental health issues.

It's reported Ms Herron and Hammond may have left a party together just hours before her death.

An online fundraising appeal to help pay for her funeral has been set up ahead of a community vigil for Ms Herron on Friday.

The Go Fund Me page set up by Melbourne Homeless Collective has exceeded its $10,000 goal.

"We're raising money to cover the costs of Courtney's funeral expenses, to save her family from further grief at having to find funds to bury their daughter," organisers wrote online.

Ms Herron's death is the fourth killing of a young woman in a public Melbourne place in the past year, including the June rape and murder of aspiring comedian Eurydice Dixon, 22, in a neighbouring suburb.

The vigil will be held on Friday evening at the park where she was found.

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Australian Associated Press