Okay this is a easy guide on how to get speech to 100 in Skyrim!
Okay to start what you need to do is go to Riften. Once you are in Riften,
Go straight across that bridge and take a left. Once your in the main bit of Riften with the circle center-
look right and you should see a place called the Black Briar Meadery.
Go on in there and you should see a man behind his counter called Ungrien.
Start a conversation with him and choose the option- "Tell me about Mavian Black-Briar." You will then get other options.. Depending on what your speech is already you will be able to persuade him. But if you have a low speech level he will just say "That is the truth my friend". So if that's you, you HAVE to give 28 gold to get the truth. Until your speech level gets to 25 will you be able to persuade him for the truth.
Whenever you get the ability to persuade him, just keep rinsing and repeating. Keep choosing the "Tell me about Mavian Black-Briar", then the persuade option. Just keep doing that and it should get your speech to 100!
Enjoy having amazing speech skills now! This ability does come in handy because you can persuade your way out of nearly anything!