I can recommend looking into the IPCC report, chapter 9, Evaluation of Climate Models [0], chapter 11, Near-term Climate Change: Projections and Predictability [1] and chapter 12, Long-term Climate Change: Projections, Commitments and Irreversibility [2].

The models run on inputs and include uncertainties (how much emissions will there be? How many clouds will really be produced in a warmer climate? ..), thus you get a range of results. The first step is to reproduce historical data, then you run the model into the future. The best prediction is given by the aggregate of a lot of different models with a realistic range of inputs.

For numbers check chapter <11.3.2 Near-term Projected Changes in the Atmosphere
and Land Surface> and chapter <12.4.3 - Changes in Temperature and Energy Budget>. Of course we're predicting climate here, not weather, so the exact temperature in exactly 100 years is unknown.

The wikipedia article is good for a start on the math: https://en.wikipedia.org/wiki/Climate_models
If you have some basic knowledge of math and programming you can create a zero-dimensional model yourself. There are probably a lot of ressources out there to help with this too. The physical knowledge needed is basic radiative things such as the Stephan-Boltzmann law. [3] (https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law)

(This will inform you about the effects of albedo changes (ice and clouds are bright and the ocean is dark, forest is a lot darker than fields and cities), changes in solar irradation and Earth thermal emissions (changed by e.g. cloud cover, aerosols such as dust and greenhouse gases.)

You can run a full model yourself, but that easily turns into a full-time job. Here's one, for example: [3]
If you want to check the ones in the IPCC reports, here's an overview: [4].
Not all of them are available, sadly, but there is a general push to make it all open. (Many scientists are ashamed of their horrible coding practices.. but these things are better than most because they need to be high-performing. FORTRAN is quite popular..)

Whether you need or want HPC really depends on the resolution you want to model at and what to include.

Do you want local and regional resolution? What vertical resolution do you want? How long into the future do you want it to run? How many different parameter inputs do you want to check?

(In the IPCC reports you can find a couple of standard emission timelines for the future, for example. The 'business as usual' one usually leads to pretty horrible results. The 'stop all emissions right now' one seems to lead to manageable results, only the coral reefs and glaciers are in danger then.)

How many feedback loops do you want to include? Do you want to include the full ocean as well? How about other things such as the carbon cycle?

(Naturally CO2 is emitted from volcanoes, changes state during rock weathering and is then eroded into the ocean, sequestrated into the sediment and subducted back into the mantle, then again emitted from volcanoes. This process is really slow though.)

Do you want to check for secondary effects such as sea-level change, ocean acidification, changes in precipitation pattern, changes in cloud cover? (which are feedback for your model) It's best to estimate the largest possible effect of each of these and then start with the main one and keep refining, which is the current process. You can follow this if you read the IPCC reports of different years. Chapter 12 tells you more about the recent refinements.

If you write emails to the authors of the report and papers with questions you might just get an answer btw, as long as you seem willing to understand and listen. It is exhausting to be called a liar when you and your peers do your very best to get it right and it still looks horrible.

[0] https://ipcc.ch/pdf/assessment-report/ar5/wg1/WG1AR5_Chapter...

[1] https://ipcc.ch/pdf/assessment-report/ar5/wg1/WG1AR5_Chapter...

[2] https://ipcc.ch/pdf/assessment-report/ar5/wg1/WG1AR5_Chapter...

The overview is here: https://ipcc.ch/report/ar5/wg1/ The summary chapters are good as well.

[3] https://opensource.gsfc.nasa.gov/projects/GEOS-5/

[4] http://www.easterbrook.ca/steve/2009/06/getting-the-source-c...