When building a model, it is always important to identify what you know. Then we can try to identify what physics need to be included in the model. Based upon the comments above, this is what I gathered:

- You have a volume that is filled with water. Do we know the temperature and flow rate of the incoming water?
- Inside we are cooling a silicon wafer (or heating some water if you want to take that perspective). Do we know the surface temperature of the wafer (this might be the information you are trying to get from the model)? Or are we going to give it a certain heat generation? What do we know here?
- What is a rough estimate of the maximum temperature of the fluid? Yes this will be output of the model. But if the temperature never exceeds the boiling temprature, we can avoid using a costly VOF model.

For your questions:
-  I understand you're not meant to mesh the container normally for fluent problems however I would like to see the chambers temperature change after the water heats up from the silicon wafer. 
Do you mean the glass container which holds water? Do you expect there to be significant heat transfer into this body? If you want to see the temperature inside this solid body, then we will have to model it. But if we are interested in what is the temperature of the water and the primary heat sink is water leaving the domain, then we can leave it out.

- Is it possible to do so and is it too overly complicated that I should just focus on the temperature change of the water (and not mesh the container)?

We can model a lot with ANSYS. Deciding on how to build a model begins by identifying what we know, the relevant physics and the desired outputs of the model. Then we have to take into account the computational resources we have and the deadline we have to complete the project. We are still gathering so we cannot say.

- I'm sturggling to see the convective heat transfer between the silicon wafer and the wafer. 

Fluent does not differentiate between convective & conductive (well except when we are specifying the boundary conditions). It is solving the energy equation. What output are you using to see the convective heat transfer?