Maybe 'unrelated (elder) brother'/没有亲缘关系的哥哥 or 'non-relative (elder) brother'/非亲属的哥哥.The second one is created by myself.

You can say: "I just call him my brother, he's not my real (or biological) brother." Is he your cousin? Chinese people have a habit of calling their cousins "brother" or "sister". It makes foreigners very confused :-)