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Step Number 1
Go to a popular search engine, like Google.com. And search for their full name inside quotation marks. The quotation marks will make your search yield more accurate results. You also want to search for their other details, such as social security number etc. You may find what you're looking for. You need to lookout for newspaper's published arrest articles. This can get quite time consuming if the person has a common name, or if you don't have enough information.
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Step Number 2
You can also get in touch with the Federal Bureau of Prisons'. They have an inmate locater service, that can be quite useful to people who are looking to perform criminal background checks. You should also check with the national sex offender registry... they'll be able to tell you if a certain person has committed a sexual crime.
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Step Number 3
If none of the above works out for you, then your only option is to sign up with a professional criminal background service. There are a wealth of them on the Internet and they all work fairly well indeed. The majority of these websites charge users over $100 for access, but I have seen a couple of these background search websites charge under $40.
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security
With this information, it should be obvious what you should try to do when it comes to looking up criminal backgrounds of people. The quickest and safest way to find out information is to pay a fee for security. Be sure you know where the information is coming from before you take it for truth.