Well, first, what you have are integer values of 1,2,3,4 with equal probability of being selected. As such, you have a probability mass function that is referred to as a Discrete Uniform Distribution.

You are correct regarding that it is the CLT that makes the distribution of the sum of the n=5 trials for the N=100 students more "normal-like." For example, if n=1 trial for each of the 100 students, then the distribution would be Discrete Uniform. A Discrete Uniform distribution has a population mean of (a + b)/2. In your case, (1 + 4)/2 = 2.5.

In terms of the (Strong) Law of Large Numbers, you need to repeat your sampling with N=100 new students. It is most likely best that you just compute the mean (i.e. Sum/5). So repeat the experiment, T times, and compute the mean on your T samples with N fixed at 100. So, you would have XBar1, XBar2,...,XBarT sample means that are going to vary from the population mean of 2.5 because of random sampling fluctuation.

Next, compute the "mean of means" i.e., (XBar1 + XBar2 + ... + XBart)/T. This overall mean will be approximately equal to the population parameter of 2.5.

Theoretically, you would need to push T to infinity and then the "mean of means" will converge to the population mean of 2.5. This is basically what the Strong Law of Large Numbers is about - without going into to more detail.