It depends on the circumstances of the murder. If someone is killed in circumstances which mean there was a high probability that they'd had no previous contact with their killer and their personal identity and previous life can have no connection with their death - e.g. in a robbery - then the police would probably not waste a great deal of time investigating the victim's background.
However, it's a fact that the majority of murder victims know their killers, so, in "normal" killings, where the murderer obviously had some kind of motive for the act, a reason for killing the victim, the police will investigate the life and behaviour of the victim. This could easily include a search of the victim's home, where they would be looking for evidence of connections to possible suspects. As jclarkdawe says, "the police always have a basis for their actions." Put yourself in their place. Ask, how could a search of the victim's home perhaps progress the enquiry; how likely is it that such a search could shed some light upon the identity of the murderer and/or the motive for the killing and the reason why this particular person was chosen as the victim? If there's a possibility that it could shed light on the case, you'll search. Otherwise a perfunctory examination of the victim's home for the sake of completeness will suffice.
As far as the mechanics of such a search are concerned. Many murders are committed by people with a close, often familial connection to the victim. Bearing that in mind, the police would conduct such a search fairly soon after the discovery of the crime, in order to ensure that any potential evidence wasn't removed, or tampered with. They would remove any potential evidence, or potential source of information - e.g. computers, personal papers - for further investigation, or forensic analysis. In the UK, they would list such items and leave a written reciept with the family, so that they can reclaim the items after the end of the investigation.