What is my best way to get a divorce if my wife and I have not lived together for 5 years?

There is a legal remedy known as "divorce by publication" that would cover a situation such as this. In legal proceedings the concept of "notice" is essential. Judicial action should not be taken against someone without giving them opportunity to appear and explain their side of things (in this case, to answer the divorce complaint). Therefore, even if a spouse can't be located they still must be notified of the divorce action before it can proceed. This is what a "divorce by publication" accomplishes.

In brief it works like this, the filing spouse (i.e. Petitioner) must make a good faith effort to find their missing spouse (i.e. Respondent).  They will have to present proof to the court that they made every effort to locate the other spouse. To prove this the Petitioner will have to show the court that they checked with family/friends, voting records, the phone book, DMV, and any other source that would likely lead to uncovering the Respondent's current whereabouts.  Once it has been demonstrated to the the court's satisfaction that the Petitioner has made a diligent search, they will be allowed to serve the Respondent by publishing notice of the divorce in a newspaper (as opposed to the more usual method of personal service).  The court will instruct as to which paper should be used (typically one in the area of missing spouse's last known address).  

In most states, the Respondent has 30-60 days to file a reply after the first day of publication. If they fail to respond within that time, the Petitioner then files a request to enter a default dissolution of marriage. It is generally granted upon the terms requested by the Petitioner (although the Respondent is given a certain time limit in which they can appeal).