I had a post on my first technology blog that looked at all the different options for how you can chat on Twitter. I have played with a lot of different options, and others will have their way of doing it. This is the way I like the most.
Either site essentially takes the hashtag and creates a chatroom for you. There is an area for you to write your message. The site automatically adds the hashtag to the end of your message. You can retweet and favorite from the site. It ultimately removes all the clutter from a Twitter feed while continually updating you on the new tweets with the hashtag.
I use this for not only my tweet chats, but also when I attend conferences and events to follow and participate in the "back channel". The only other suggestion I would recommend is to have your Twitter open in another tab to keep track of people responding to you.
One thing to note - while not impossible, you can do this on a mobile device. A keyboard makes it so much more efficient.
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