A Police Clearance Certificate, also known as a Criminal Background Check, Criminal Record Check, Certificate of Good Conduct, or in the case for Americans, an FBI Identity History Summary, is a document detailing an individual's criminal history. It is commonly required as part of a visa and/or citizenship application.

If you have lived longer than 180 days in countries outside the US, you are typically required to get background checks for those countries as well.

For your US background check, you will just need to get an FBI Identity History Summary, including an Apostille of the check and in some cases a sworn/certified Translation.

Getting an FBI Identity History Summary

The general process requires an applicant to take or have their fingerprints taken, sent off to the FBI, and then verified by an agent, who will then have a complete background check sent back to the application. The entire process can take 2 to 3 business days for expedited checks, and upwards of 30 days for regular checks in busy times, or if the applicant made an error (such as invalid fingerprints).

Once a background check has been completed, the applicant will typically only be given 60 to 90 days by a foreign country to use it. This can result in an extremely stressful application process, especially when an apostille is required as well (which have their own processing time). For this reason, if you are working on this process from abroad, it is HEAVILY recommended to seek the services of a "channeling" agency. They are able to retrieve and manage documents and communication on your behalf, and typically have much more direct lines of communication with the FBI and Office of Authentication.

The procedure does not include storage of the fingerprints nor reference to FBI or other federal fingerprint databases.

There are two primary methods to request a background check: with aid from an authorized agent (such as a police officer or post office clerk), or manually by the individual. The former can only be performed within the US or by recognized foreign companies (which DOES NOT include any US government operations), and the latter can be performed anywhere in the world with you, a printer, fingerprinting ink, and possibly a friend to roll your fingers for you.

Assisted:

This is the easiest route and you can typically get your results emailed to you in a secure PDF within 48 hours. You can also expedite a hard copy by express mail (though this is rarely necessary).

USPS: In order to obtain a background check with the help of a USPS location, you will need to register your information online and find the post office most convenient for you which has fingerprinting service. When you're there, they will take your fingerprints digitally, and send you a background check by e-mail (and by mail if requested).
FBI-Approved Channeler: With this option you visit a private business that has an FBI terminal for taking fingerprints. These can be more convenient options than USPS but also cost more. See the FBI's list of approved channelers or search for fingerprinting in your area.
Local Police: The website National Background Information contains a large and fairly complete list of police locations, state-by-state, which offer fingerprinting services. Some may take inked fingerprints, while others may offer digital services. If the former, there is no guarantee they will also be able to send your check to the FBI, so you may still need further assistance. It should go without saying that caution must be exercised going to any police office as a marginalized person.

Manual:

In order to manually acquire a background check, you will need a few supplies:

Form FD-1164: Instructions for filling out this form (specifically FD-258, but they are VERY similar) may be found here.
Fingerprinting Ink: Theoretically, any ink CAN work, but you will want a fingerprinting pad made specifically for this purpose.
...and a lot of patience: Fingerprinting can be extremely frustrating to do, and you'll likely want at least five sheets to account for making errors you don't catch. A guide such as this one is a life-saver.

Once you've captured your fingerprints multiple times on multiple forms (or had the police do it for you), you can send them to the FBI. You may send them directly using the information on their site, or you may choose to employ a channeler who can manage the process for you. These people may also accept Form FD-258 rather than 1164, which is a near identical fingerprinting form. They may also have their own offices near you in the US which can take fingerprints. If you are doing this from abroad, YOU ALMOST CERTAINLY NEED A CHANNELER DUE TO POSSIBLE CHALLENGES WITH INTERNATIONAL MAILING AND COMMUNICATION.

With any luck, you will have a background check sent to your email within a week once it's complete.

Obtaining an apostille

After you have obtained your FBI background check, it is very likely you will need to Apostille it for use in the country with which you are requesting a visa or residency from. The two methods for doing so are to hire the help of a channeler, or to do so manually. Regardless of which route you choose, you can use a printout of the PDF you receive by email to be apostilled. This is because even the hard copy you may elect to receive is just a printout of a digital seal/signature.

Channeling:

Monument Visa: For $75 with standard service or $125 with rush service, Monument Visa will manage the apostille process. You will need to provide them with the digital copy of your FBI Background Check, relevant personal information, and your mailing address (foreign accepted). They will accept payment by credit and debit card. While they offer translation services, you MUST verify through your own research if the country you are applying to will accept a translation performed and notarized by them. Not all countries will.

Manual:

Only the Office of Authentication is authorized to apostille documents, and requires a few items from the requester to do so. Up to date information can always be found on the US government website.

Original Document Copies: The ORIGINAL copies of your documents. No photocopies.
Form DS-4194: This form will require you to fill out your personal information, shipping information of your documents, types of document, and projected cost.
Money Orders or Checks: You will either need to obtain money orders from your bank, or provide checks. It is unclear if these must be per document or if a single payment for the full order is sufficient. Due to international banking difficulties, this is reason alone NOT to do this process manually abroad.
Self-Addressed Pre-Paid Envelope: You must include a pre-paid envelope in which everything can be returned to you by the government. Foreign addresses are, seemingly, accepted.

Obtaining a translation

For countries that will not accept English-language documents, you will need to have your apostilled background check translated into the language of the country you desire to use it in. While some countries may accept translations performed within the US, many will not. This information should be detailed in individual country pages, as it is very specific to each countries legal and civil systems.