I finally became to like the subject. 是错误的。
正确的句子应该是：I finally began to like the subject. 或者：I finally came to like the subject.
2008年7月第1版《外研社英汉多功能词典》第177-178页指出：表示“开始变得...”不用 become, 而用 come to do：
Somehow I've come to like Ian. 不知为什么，我开始喜欢伊恩。