Edit: Oh sorry, I think I misread: You want to model the likelihood that a particular user is logged in at a particular time, i.e. given a certain user and time of the day you want to receive a likelihood that this user will be logged on at that time of the day?
Is it the case that you want to treat each user individually or do you want to get a general feeling about the time when users are logged on?
Or is the nature of the question more that you want to figure out whether or not a certain user is a 'morning-user' or an 'evening-user'?
I thougth you just want a likelihood for the log on time...
--- old answer ---
For now it seems as if you are just interested in the time of day, i.e. you want to ignore the fact that there is a summer / winter period and there may be differences depending on the day in the week (more logins on the weekend?). Hence, you could just turn timestamps into 'how many seconds after midnight', i.e.
'2000-01-01 01:00:00' --> 3600
'2016-05-03 02:10:23' --> 3600 + 3600 + 600 + 23 = 7823
Then you treat these numbers as continuous values. Then you could theoretically fit a GMM. However, it has hyperparameters which you need to optimize. Since you are lucky and the data is just one-dimensional, you can fit different GMMs (i.e. GMMs with different amounts of clusters) and see how good they resemble the data 'by eye'. Once you are satisfied, you got the likelihood.
However, depending on the question this can be useful and 'unuseful': GMMs will cluster the different logons into groups along the day and it will likely figure out something like 'there is a group in the morning and another one in the evening' or so. I.e. the latent variable, GMMs will uncover is the number of the cluster. However, if you are just interested some kind of likelihood then a one dimensional density estimation will suffice...