If I'd have a clone of myself (with my DNA) and I would record the development of that person, would I be able see exactly the same development as myself (same personality, appearance)? For example that at a certain time, that clone would look how I looked like when I was 8?
No, your clone would differ from you. Actually such clones exist in humans also. Monozygotic twins are clones. They certainly look alike but they definitely differ in some respects.
Understanding why this is true require one to understand the concept of heritability from the field of quantitative genetics. You should definitely have a look at the post Why is a heritability coefficient not an index of how "genetic" something is?. The main reason why you would differ is because our phenotype (personality, appearance, etc...) also depends on the environment we experience.