It depends on what type of claims you want to handle.
Some companies only hire people with no insurance background as they want to train them their own way and don't want anybody who's not new to the business. Other companies do the opposite and only hire experienced people. Some hire both.
If you want to do auto claims start in the front office at a body shop or Enterprise Rent-A-Car. Spend a year or two and then make the jump.
If you want to do property claims try to find a smoke/water damage restoration company or other type of contractor to work for and then make the jump.
If you want to do injury/liability claims it helps to have worked in a law office or have entry level experience doing medical claims data entry or something like that. Minor business experience is sometimes adequate for entry-level jobs.
More sophisticated types of adjusting such as commercial liability, construction defect, surety, workers compensation, and litigated high-level liability claims usually require a number of years doing adjusting in one of the above fields first.
All of these jobs require you to be able to do critical thinking, have basic computer skills, and some level of sales & negotiation ability (except the data-entry express claims jobs). Simply being bi-lingual is a huge plus with many companies as they never seem to be able to find enough of these people.
Field claims jobs that require face-to-face interactions with customers and agents also require conservative appearance (ie. no freaky tattoos or piercings you can't cover with a shirt, etc...)
All of these jobs at their entry level may sometimes be obtained with very little experience. Claims adjuster's "first careers" cover a wide variety of positions. One person I know was an elementary school audiologist, one was a manager at Pizza Hut.
At the very highest levels of liability adjusting, where salaries approach and sometimes exceed six figures, one needs a level of knowledge of tort claims law that can exceed that of the lawyers they hire to litigate the claims. "Balls-of-steel" to make decisions on claims that potentially could cost millions of dollars or cost the person their job if they're mishandled are also a plus.
Independent property adjusters who handle disaster claims year-round can make several hundred thousand dollars a year in busy storm years. This requires an almost insane level of organization, dedication to leave home on a few hours notice for storm duty that can last for months at a time, ability to handle working in conditions ranging from adverse weather and lack of basic services like electricity and communications to being in areas that are unsafe due to structural damages or an absence of adequate law enforcement resources.
You may or may not need a license to be a claims adjuster or motor vehicle damage appraiser. For example in Connecticut you need a license to do either. In Massachusetts you only need the license to be a vehicle damage appraiser but not to be a claims adjuster. In Kentucky you only need a license to be an adjuster but not to be a damage appraiser. In Ohio you don't need a license to do either. In some states you need a license specifically for the type of damage you are looking at. For example in California you need a special license to handle appraisals on earthquake claims. You usually do not need one to work as a repairer of houses or cars no matter where you are. The licensing test is not difficult if you study-up a little beforehand or have any experience.
A large percentage of available openings in this industry are posted in October, November, and December as this is the period when claims departments usually receive their budget allocations for the following year and find out how many new people they can hire.
2011 will likely be a thinner year than normal for hiring in this industry as property & casualty companies are currently suffering massive losses due to the unusually large number of storms this year combined with the unusual number that hit heavily populated areas.