in the last semester I learn homological algebra and higher category theory\homotopy theory

but I am kind of confused when I try to realy understate the link between the two subject ( this is really not my comfort zone ...)

Therefore I try to write(a kind of self exersice) a text on homological algebra and homotopy theory but starting at a low level (accesible for students which do basis courses in algebaic topology , commutative algebra and category theory )

I would like to introduce the following concepts in homological algebra :

1. chain complex

1$\frac{1}{2}$.grothendieck group

2. homotopy of complex

3. derived category

4. t-structures

And also I would like to introduce the following concepts in homotopy theory :

1. Model categories

2. Homotopy category of a model category

3. Derivation in the setting of model categories

4. Quasi categories

4.5. simplicial object in a category and homotopy in this context

5. Dold kan equivalence

Now the "hard" part start :

how to organize this concept in a good way? for 1-3 (either in homology\homotopy) I think that I know how to do that but for 3-5 especially in homotopy I dont have any idea ...

This gives rise to my questions :

1. how to give motivation infinite category or with more generallity homotopy theory\higher category theory but from a homological point of view I read somewhere a maybe good idea :

For an abelian category $\mathcal{A}$

the derived category $\mathcal{D(A)}$ is not defined by a universal property

I read somewhere that in some sense higher category theory resolve the problem,Okay but why ? and do we need quasi categories or model categories will be sufficient for doing that?

2. and if someone have some idea to organize this text I open to any suggestion

I will be grateful if someone could give me some clues for doing this self exercise .