Q:

How is a sentence constructed in German where the direct object takes an action? 

For example, if in English I say 
"That makes me feel better"
"Me" is the direct object as it relates to "makes," but then it is the subject as it relates to "feel."  I don't know if there is a grammatical term for this kind of sentence.  
I can't figure out how this should work in German, and don't really know the right search terms to figure it out by googling.  The direct way would be
"Das macht mich besser fühlen" 
But that really doesn't seem right to me.  I've tried typing in forms of this sentence into dict.cc to see if I can find a similar phrase, and everything seems to come up with other ways of constructing the sentence that get around this.  Perhaps it is just not a valid construction in German?  Maybe this sentence would need to be translated
"Deshalb fühle ich besser"

A:

Your problem is the verb make.  That one is special in English as it takes another verb in infinitive to express the result of an action. 
In German, a similar construction with bringen plus noun made from an infinitive can be used:

Er bringt mich zum Lachen. 

He makes me laugh. 
Even in cases when it sounds a bit awkward in English:

Das bringt mich zum Nachdenken. 

That makes me think over it. 
However, your example would require a verb "besserfühlen" which does not exist.  You could create it on-the-fly, but it's much better to rephrase without the infinitive. 

Darum/Dadurch/Daher/Deshalb/Deswegen fühle ich mich besser. ( note the mich - me)

That makes me feel better. 

A:

English Infinitive Construction  Normally Expressed in German by Dependent Clauses
A few things complicate the issue. " feel" falls within the category of copula verbs (no objects).  It is a way of equating the subject to some other thing.  You can express that in German without the explicit infinitive verb "feel". 

That makes me feel better. 

Das tut mir gut. ( That does the body good lol)
Das macht mich glücklicher. ( That makes me happier)
Das beruhigt mich. ( That reassures me. " That makes me feel better" usually refers to emotional status, not to other issues, which is why I'd favor this translation)

Consider the following case though:

I want you to tell me the truth. 

German does not support infinitive constructions like "to tell" after another clause, you're right about that.  So we add a dependent clause:

Ich will, dass du mir die Wahrheit sagst. 

I want, that you tell me the truth. 

Additional example

I believe her to be honest. 

Ich glaube, dass sie ehrlich ist.