How is it decided where 了 goes in 这时他得到了公众的支持? In the Chinese wikipedia entry for Julius Caesar (the play), there is this sentence: 


  凯撒死后，布鲁塔斯进行演讲为自己辩护，这时他得到了公众的支持


Firstly, 进行演讲 means "to make a speech"/"to carry out a speech". Why is 了 not placed after the verb 进行? I suspect it is because the sentence is not finished and the tendency is to place 了 after the final verb in the sentence (in this case, 得到), but please tell me if I'm wrong.

Secondly, I don't understand when 了 is meant to come after the verb or at the end of the sentence. E.g. what is the difference between 这时他得到了公众的支持 and 这时他得到公众的支持了?
My take on this is that, as you say, the aspectual 了 is placed after the final verb of the sentence, but I'm not aware of the existence of a hard rule for that. I would say it is mainly a matter of style. Maybe there are linguists here who can confirm this.

As for how to use 了 and where to position it in a sentence, this has been asked many times and you can find some pretty good and accepted answers here:
Tense and use of 了


  what is the difference between 这时他得到了公众的支持 and 这时他得到公众的支持了


The first one 这时他得到了公众的支持  is describing a completed action and sets the time in the past --> That time he obtained the support of the crowd.

The second one 这时他得到公众的支持了 describes a change of state (and tells nothing about time) --> This time he's got the support of the crowd. 
Which implies that he usually doesn't have it. 

However, since this last one example describes a change, the sentence with 得到 - which I perceive as a punctual event - plus the absence of a context sounds a bit weird to me.

So I'd prefer the first version any way.