Some chats are quite structured with a set topic and/or a host asking specific questions, and rules about tweeting links etc. - try and check out the rules and follow them! Winding people up isn't gonna get you anywhere and chats are more enjoyable for everyone when people stick to the rules.
Join in, respond to people
There's no good being a "lurker" in a Twitter chat! If someone asks a question - answer it. If someone says something you agree with, say so! If you disagree, say so (politely of course)! Get involved as much as you can.
There's no better way of starting conversations than asking questions! It might be something about the chat, for example what the topic is or who the host is, or something related to the topic - you might ask about people's top recommendations from LUSH in a chat about pampering, or if people have any tips for organisation in a more general chat. Make sure you respond to anyone who answers your questions, try to keep the conversation going!
Make sure you use the appropriate hashtag in all your tweets, or people simply won't see them! And don't be afraid of butting into conversations that look interesting - that's the beauty of Twitter chats. You get dozens of little "side-chats" springing up as a handful of people here and there find a topic to discuss.
Be helpful, answer questions
If you see someone ask a question you know the answer to, then answer it - for example people who are late might ask who the host is or what the topic is. If someone's asking something else that you know about, say so, and if you have a blog post about it - offer to link them!
If you ask for links, try to check at least some of them out & leave a comment saying you've come via the chat
It's common at the end of chats to ask for links to people's blogs - I usually send a tweet out towards the end, sharing my own link and asking for links in return. If you do this, please try to have a look at the links you receive and, if you can't leave a comment for whatever reason, at least tweet the sender with some feedback - even if it's just "Love your header!" It's not nice to feel ignored.
I have a tendency to tweet things about how quickly the chat is going or how I'm struggling to keep up (I always struggle to keep up) - things people will identify with. These pretty much always get a few responses. I also think it's polite to to tweet the host (if there is one!) at the end of the chat to thank them.
If you get chatting to someone, follow them!
An obvious one, really, but if you've exchanged a few tweets with someone, it's nice to follow them so you can carry on chatting to them in future.
This is the best way I've found of keeping up with chats, as they can move very quickly, with hundreds of tweets being sent per minute. I generally have one tab open with my notifications so I can see if people reply to my tweets, one open with the hashtag (set to 'live') so I can see all the tweets as they happen, and one of the host's profile (if there is one) so I can see any questions etc. You can do this through Tweetdeck, too, but I just prefer the desktop version of Twitter.
Have a look through your Twitter Analytics afterwards
This is mostly just because I'm a bit of a geek and I love stats and analytics of any description, but I always like to have a little look at my Analytics after a chat - I find it so interesting to see how my impressions and engagements shoot up on days when I've got involved with a chat! Hopefully you can see below that I got a good couple of thousand more impressions on the 10th and 24th of April when I took part in the #bbloggers chats.