In practice, you want some arbitrary precision arithmetic (a.k.a. bigint or bignum) library. My recommendation is GMPlib but there are other ones.
Don’t try to code your own bignum library. Efficient & clever algorithms exist, but they are unintuitive and difficult to grasp (you can find entire books devoted to that question). In addition, existing libraries like GMPlib are taking advantage of specific machine instructions (e.g. ADC -add with carry) that a standard C compiler won’t emit (from pure C code).
If this is a homework and you are not allowed to use external code, consider for example representing a number in base or radix 1000000000 (one billion) and code yourself the operations in a very naive way, similar to what you have learned as a kid. But be aware that more efficient algorithms exist (and that real bignum libraries are using them).
A number could be represented in base 1000000000 by having an array of unsigned, each being a “digit” of base 1000000000. So you need to manage arrays (probably heap allocated, using malloc) and their length.