there are many ways, basically you post on twitter the link of the website you want to increase its traffic. And there are other sophisticated ways, like customizing your background to include information like the website url, blog url, business email ... etc, but bear in mind they will not be clickable. Also, this technique requires you to be familiar with photoshop or a similar software to prepare such background with the exact dimentiones of your twitter page.
Good Luck !
Also, I recommend you to Google "twitter backgrounds" and search for a few Twitter background tools that dont require photoshop. Remember with twitter - you need to build a following as well - by outreach and following other Twitter users.
Putting twitter icon your website and more people will follow you in twitter when you write something useful or valuable. You will gain more traffic as your followers who are getting more and more. Do note that spamming followers are not very useful and you should try to have more followers that sincerely do want to read your tweets.
If you want to get traffic from your twitter, use keywords on your posts, give link at your post, try to follow others , increase your followers, before the keyword use this sign"#".Also add your twitter button at your site.
This is might help you.
Intreresting tweets get you more followers. Send thank you message to every follower and keep sending direct messages, replies and retweets. As your fololowers increase, you will get more traffic to your site from twitters.
Post interesting things on twitter. Retweet interesting stories from other sites. Put a Tweet Me button on content on your site (use tweetmeme.com) and retweet your own stories too. Engage your followers and be sure to do a combination of retweeting stories and posting unique content.
There really is no insider tricjk to getting traffic from twitter, especially if you want traffic that converts. Find like minded people by doing a twitter search for your keywords, make friends and then interact with them
I think autofollow is a waste of time. Building real relationships is the way to go. No point in having 10,000 followers if none of them is actually interested in you
Engage twitter with a beginner’s mind. Give what you want to receive by helping and praising others. You will get help and praise in return. Also, focus on adding useful content and keep a positive state of mind.