Miss Lawrence, 35, was last seen five weeks ago near her home after she finished work at the University of York's Goodricke College.
At a news conference in York, Detective Superintendent Ray Galloway said: "At the moment we have no proof of Claudia's death. However, five weeks on from her disappearance we also have no proof of her life.
"The investigation is now being formally classified as one of suspected murder."
Mr Galloway said he had not given up hope but was "looking at this investigation from a technical perspective".
Miss Lawrence's father, Peter Lawrence, who was at the news conference, said he understood why the police were now treating his daughter's disappearance as murder but said he believed she was still alive.
He said: "It is necessary for their investigation but we believe that Claudia is still alive and that is what keeps us going and motivates us.
"I believe and the family believe very much that Claudia is still alive and we very much hope she is alive, that is what is keeping us going here.
"The sense of hope is clearly there."
Mr Galloway also disclosed new "significant" lines of inquiry after information from members of the public.
He said one person reported seeing a man and a woman walking near the University of York between 6am and 6.30am on 17 March and 19 March.
Another witness also saw a man and a woman at 5.35am on 19 March as he travelled down Melrose Gate, towards Heworth, where Ms Lawrence lives.
Mr Galloway said there was no specific interaction between the couple but said the man was wearing a dark hooded top with the hood up and was smoking a cigarette with his left hand, which was "unusual".
Mr Lawrence made another "direct appeal" to his daughter today.
He said messages had been left on her mobile phone this morning urging her to get in touch.
He said it was "so unusual" for her not to be in contact and, despite many appeals, there was no word from her and the family was "extremely worried".
He said: "Your mum and I have this morning each left a message on your mobile phone just in case you are able still to listen to messages.
"Please, please, listen to these messages and, if you able to, please send just a quick text or make a quick call.
"You may not be able to do this, but we just hope there's a slight possibility that you can.
"There's absolutely nothing for you to worry about in terms of getting in contact.
"You are not in trouble with anybody. We just miss you. We just want to know that you are safe."
The man seen on 19 March on Melrose Gate was also wearing dark combat trousers and was said to be skinny.
The woman was described as having mousy brown hair and wearing a blue jacket.
They were seen on a bridge near an electricity sub station on Miss Lawrence's route to work.
Mr Galloway said: "We don't know it is Claudia but it is a significant sighting at the right time in the right location."
He said the other man seen on University Road was described as aged between 50 and 55, with dark brown hair, greying at the sides and a dark jacket with lighter coloured trousers.
He said the couple may be the same people who were seen by another witness arguing by the side of the road near an open car on 19 March.
Mr Galloway urged both couples to get in touch.
He also appealed for university students returning after their Easter break to come forward if they remembered any information which would help with the investigation.
Crimestoppers is putting up a £10,000 reward in connection with the investigation.
Dave Hunter, Crimestoppers regional manager for the North East and North Yorkshire area, said: "Crimestoppers has put in an enhanced reward of £10,000 for information relating to this case.
"It is quite unusual because, at this moment in time, no crime has been committed. However, it is such a big issue and the public have been captivated by the case.
"From a Crimestoppers point of view, we hope this will generate specific information relating to Claudia, which can help police with their inquiries."
Miss Lawrence, 35, was last seen on 18 March after she finished work at the University of York's Goodricke College and has not been in contact with anybody since around 8.30pm that evening.
She was due to start work at 6am on 19 March but never arrived.
Police have said they fear she may have come to harm after meeting up with someone she knew.
The investigation into Miss Lawrence's disappearance is the biggest the North Yorkshire force has conducted since the search for multiple killer Mark Hobson five years ago.
