An obituary can be the key to solving a family tree mystery.
It is absolutely necessary to understand that NOT all obituaries and death notices are online.   Anything that is online is there because someone paid for internet access and webspace. They took the time and energy to code it and post it. This is highly time-consuming, more so with graphics, those handy search calendars and boxes for surnames, the interlinked pages, etc.   There is no magical, mystical, instant method of taking printed material and turning it into webpages.   Keep your expectations reasonable, and realize that YOU will have to do some work.
If you know where the deceased died, a telephone and address search (for the same surname) in the area he / she was known to live will provide a list of names to whom you should write with your request for information regarding the deceased.   Identify yourself and your connection.   You will not receive replies from even a quarter of the recipients, but those you do get can pay off handsomely.   Once you've exhausted the leads from this list, spread your search over a wider area.
Placing a letter to the editor request for information can be quite helpful.   Two letters in two papers in two states yielded four correspondents and much useful information for me a few years ago.   All were in some way connected to at least one of the names I listed in my letters.   One was quite hostile and rejected my requests, yet I was still able to glean from her brief note some information and the happy fact that a family tree had been completed.   This I later found online and as part of a genealogy programme's CD collection of family trees.   I may not have looked otherwise.   Another sent me an obit clipping she'd had for nearly ten years.
The first step is to determine when and where the deceased died.   In the United States and some Canadian provinces (i.e. British Columbia), death registries are kept and are searchable online.   Those requesting copies of official documents may have to prove some familial connection. A form that must be properly completed and fee is almost always involved. There may be limitations on releasing such information. Death records may not be available for several years or even decades, except to immediate family.
The death registries initial search may include the last place of residence, or in the case of the American SSDI (Social Security Death Index), the place where the death benefit was sent.   One can reasonably guess that a surviving spouse or child lives in this area, or at least received mail there.
Armed with the location, begin a systematic search for:
newspapers
university and public libraries
funeral homes
hospitals, nursing and chronic care homes
churches (if applicable)
local coroner's office
local historical / genealogical societies and museums
service clubs, fraternal & work-related organizations
school & alma mater societies
Use various search engines, as each one works in different ways. Some only return links to websites that have paid to be listed.   You will probably want to conduct the search one category at a time, following through the next steps before continuing with the other items.   Also conduct this search by mail and by telephone. Not every institution has a viable, up-to-date website, or the staff with the time to edit it and monitor e-mail.
A good road atlas is invaluable in this search, to help you determine if there is more than one location to check.   This is particularly applicable if the death location is rural or the last residence was rural.   Facilities like hospitals and funeral homes in towns and cities service areas larger than just the communities in which they are located.
Once you've found the newspapers, search their website, if one exists, for death and funeral notices and obituaries.   Many only maintain current announcements and few will respond to requests for old files other than news items.   For these latter articles, most will happily send copies for a fee, probably not more than $10-15.   Some will allow retrieval of anything from their archives via the internet once you have paid a subscription fee.   The options are quite variable.
For obits that are beyond the scope of the newspaper's holdings, the libraries are the next step. For instance, the Ann Arbor News (Michigan, USA) has a freely accessible 180 day archive online. However, the District library in Ann Arbor offers older obituaries for a fee, but only after you call the library and listen to someone there read you the copyright!
Many public libraries and often university or college libraries maintain microfilm /microfiche copies of local newspapers. Microfilm /fiche is often available via inter-library loan with a minimal postage fee or none at all.   You request and use the microfilm at your library.   Other libraries may keep the actual newspapers for several years.   You may still not find a notice if the deceased was not a resident, was destitute or institutionalized, or without living relatives or friends who cared for him.
This is where the funeral homes, hospitals and care facilities, etc. come into play.   With as much information as you can gather, dates, names, places, etc., contact these institutions to see if the deceased is in their files.   Briefly explain the purpose of your phone call, but keep in mind that their prime objective is not answering queries about the past. Privacy laws vary in each jurisdiction.   They may prohibit any divulgence of information, even to the point of forcing the staff to be unable to deny or verify that the deceased was indeed, ever on file.   All you can do is to seek reconsideration via the governing body.   Never abuse the person responding, as he or she is just doing their job.
Always offer to reimburse copying and mailing expenses.   With churches, it is always wise to include a small donation, say $5-10 with your request.   Of course, show gratitude for the time and effort extended, even if no information of any use results.   Be prepared to justify your request, to prove that you should be privy to such information.
The historical / genealogical societies and local museums may have recorded burials, or be actively conducting a cemetery project, and keep a collection of members' publications.   They may also have been the recipient of funeral home records, etc. when such businesses close or dispose of old files.
Historical / Genealogical societies and clubs can be of enormous help with long-distance searches and providing contacts and information to plan a trip in person.   Some have volunteers who will do lookups for you if you cannot travel there yourself.   Some may even do local research for you and take photographs of tombstones.   You should be prepared to pay their annual membership fee at least once, and to reciprocate in kind -- to provide similar assistance in your area.   As well, reimbursement for copies, etc. is only fitting.  
I've benefited many times from the efforts of people in places couldn't visit myself.   No one wanted anything in return, other than perhaps to request that I commit a similar act of kindness.   It doesn't hurt that I live in a community that is home to a large, historical cemetery and a wonderful genealogy library, both of which I enjoy perusing.   You don't have to be a professional genealogist or photographer to check a few books or take some snapshots.
Unions, veterans associations, fraternal organizations, school associations, etc. may all be of use to locate an obituary, as these often provide assistance and pallbearers for the funerals of their members. They may have visited the deceased in a hospital or nursing home.   Here again, you will likely have to provide proof that you are a relative.
Patience is a virtue is just one homily to keep in mind while conducting genealogical research.   Being persistent without being a pest is helpful in your search.   Be thorough and keep scrupulous records about your search, names of contacts, etc. You will avoid duplicating effort and save time.   Ask each person you contact if they can think of anyone or anywhere else for you to check.   Be cheerful, brief and grateful.
This page should give you ample ideas and avenues to search.   The SSDI and other death registries, newspaper links, vital records information, etc. are all listed in my Free Obits Online page, or go to the fabulous RootsWeb site to learn how to use death, tombstone and cemetery records or the comprehensive Cindy's List to find help at all levels in all directions.   Good luck!