Twitter Verified accounts are not for any random person. They are meant for the people with authority in their industry. For example, any random person opening account may not get the verified symbol.
The qualification criteria for the verified account is -
1. Authority in industry such as news, politics, sports, film celeb, published writer, business person, philanthropist.
2. Popular persona.
3. No tweet count relevance.
4. Person who is known to hold speech value (as in controversial)
The team at verification center always keeps changing criteria and for that reason many known people didn't got the verification as of yet. Twitter is very selective and the process for verification is not same for all types of user as well.
For example, for those with news and politics, simple image of them holding the twitter username can be used for verification of account.
For others the process is different but some of the common verification documents are by using support ticket. Go to the following URL: https://support.twitter.com/forms
Here you have to create a ticket and provide the following details.
1. Account name.
2. Identity proof (passport works better)
3. Real image holding twitter username banner
4. Real image passport size
5. Wikipedia page or bio page link.
6. Email or contact info.
Once you submit these details. You'll be either verified by call or email in some cases. For more popular personality, the given details alone suffice and can get the verification done.
If you have done the verification on Periscope and Vine then that verification can be cross over to the Twitter as well. You just have to create another support ticket mentioning these details.