Downloading your personal archive is easy, but you need to do a little manipulation before you can play with it. My tweets were time-stamped in UTC time (I'm not sure why - perhaps by default, perhaps because of my location settings) so I had to adjust this for time zone changes due to day-light savings and overseas trips (I didn't bother with domestic trips as I don't have an easy record of them, and they don't make too much difference - an hour here and there).
The following has a dot for every tweet I've written since the end of 2010. Take note that the x-axis is quite long (3.5 years) and the dots are quite large (bigger than a day). I haven't annotated it, but it is interesting to spot life events - the birth of my children, various periods of leave and holidays, over-tweeting during The Ashes etc. There are auto-tweets that came out at the same time each week (which I've now stopped as they're annoying). There was a definite shift in the time I rise in the morning after December 2010 when my son was born and a surge in late-night tweets after my daughter was born in 2013.
Breaking it down is a little more interesting. The following shows tweet frequency for work days and non-work days (weekends, leave) since the start of 2013. On a work day, I tweet in the main on the train. I usually catch a train around 7 or 8am in the morning and the return train around 5 or 6pm. During work hours there is a trickle through coffee breaks and lunch, and after dinner is another peak. This type of profile aligns somewhat with the findings of other social media studies (Yellow Social Media Report – 2014 - thanks @problogger), although the amount I tweet on the train is more than the norm, whilst the amount I tweet at work is less (although it is a great way to horizon scan the various fields of science in which I work, once you follow the right people).
Non-work days follow a different profile, at least until after dinner. There's a slightly later rise in the morning, dips when we would be attempting to get out of the house, a dip at an earlier dinner time and a large peak in the evening once the kids are in bed. This peak is higher than a work day, in which time I might be preparing for the next day or falling asleep on the couch. By about 10pm it is basically the same till 6am the next day.