Dukes County is on one of the islands just south of the Massachusetts coast, and it was part of New York until it was established as a Massachusetts county in 1695. Many records started being kept then but it wasn't until 1841 that the state made it a requirement. So documentation between 1695 and 1841 can be sporadic though there is a substantial amount of Dukes County public records for that time period.
For that range of years, you will have to contact the county clerk in Edgartown. You may also need to see the records at the specific town clerk's office, but the county office should be able to help with that.
For Dukes County public records after 1841 but still before 1921, the State Archive is your main source of information. Their website has searchable indexes that can help you find the material you need, and staff will do some photocopying and mailing for a small fee. If you are in Boston, you can always visit the Archives yourself and get all the copies you want at no charge.
Modern records (1921 until today) are available through the Registry of Vital Records and Statistics for the state, located in Dorchester. You can mail a request form to them but they also accept requests at their service counter, which is open Monday through Friday during usual business hours.
Dukes County Birth Records
Dukes County birth records are available to anyone who places a request, regardless of your relationship to the person on the record. If you send your application by mail, the fee is $28, but it only costs $18 if you go to the service counter (either the main Vital Records office or the local clerks). Return time is usually a month for a mailed request and you will have your birth records back while you wait if you are there in person.
Dukes County Death Records
The process for getting a Dukes County death record is the same as for a birth record, including the lack of restrictions and records fees. The application forms can be downloaded from the Office of Health and Human Services website, and one form is used for all types of records.
Dukes County Marriage Records
Marriage records are also available to anyone who makes a request, and the form is the same for birth and death records. You will have to fill in the names of both parties in the marriage and staff will search up to 10 years if you don't know the exact date (they will actually do this for the other types of records as well).
Dukes County Criminal Records
Criminal records in Massachusetts can be accessed through the Department of Criminal Justice Information Services in Chelsea. They do accept mailed-in requests, and you are permitted to get your own Dukes County criminal records or records for others depending on their crime and sentence. The DCJIS website has the details. The cost for each record request is $30 and usually takes about a week to get your documents.