C++ code to perform DNA sequence compression.

So you must only write the code. Take some tutorial to leverage for codind skills.

Consider to use classes to give your code a better architecture. It is best to split every task in a single function or class, than sum up all results in some array. This array should be typedef array of your results.

You will need to do some bit shifting and type casting somewhere in your code:

unsigned char byte = (dna1<<6) | (dna2<<4) | (dna3<<2) | (dna4<<0);
int value = (int) byte;
char c = (char) value;

Dont rely that the code is suitable for you!!!

Tip: write some test snippets to verify your code.

You didn't state any question or problem, you just pasted an assignment.
Your statement look weird to me:
- 4 and 5 are about converting a 8 bits value to a 8 bits value ???
- 6 this step is only looking for repeats of blocks of 16 and with distance multiples of 16 too, this means that repeats with distances different of 16 do not interest you.
- 7 is weird too since @ in ascii, it is also the encoding for taaa, how do you know what is what ?
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We do not do your HomeWork.
HomeWork is not set to test your skills at begging other people to do your work, it is set to make you think and to help your teacher to check your understanding of the courses you have taken and also the problems you have at applying them.
Any failure of you will help your teacher spot your weaknesses and set remedial actions.
Any failure of you will help you to learn what works and what don't, it is called 'trial and error' learning.
So, give it a try, reread your lessons and start working. If you are stuck on a specific problem, show your code and explain this exact problem, we might help.

As programmer, your job is to create algorithms that solve specific problems and you can't rely on someone else to eternally do it for you, so there is a time where you will have to learn how to. And the sooner, the better.
When you just ask for the solution, it is like trying to learn to drive a car by having someone else training.
Creating an algorithm is basically finding the maths and make necessary adaptation to fit your actual problem.

The idea of "development" is as the word suggests: "The systematic use of scientific and technical knowledge to meet specific objectives or requirements." BusinessDictionary.com[^]
That's not the same thing as "have a quick google and give up if I can't find exactly the right code".