I would like to make myself a Twitter chatbot.
Yes, there are some open-source tools that let you create your own chatbot which you can easily implement on your own website or your twitter page in your case.
For your requirement, Bot Libre is the best solution. It allows you to create your own Twitterbot that lets you manage a Twitter account, check status updates, manage followers, reply to mentions and posts, retweet important posts, tweet blog posts or an RSS feed.
As it says on their About page: Bot Libre allows you to create your own artificial intelligent chat bot, train them, and share them with others. You are free to use this website, and create your own bots for personal, commercial, or educational usages.