The phrase "to build a reputation" means to establish or create a certain image or impression of oneself based on their past actions or behavior. Full definition
Reviews and testimonials are your tools for building a reputation as a contractor. (topresume.com)
We have built a reputation on providing courteous and honest service. (shoppok.com)
You can change insurance companies any time you wish, but be careful of building a reputation of constantly jumping from one policy to another. (usinsuranceagents.com)