你给的几乎就正确了,只要稍微改动就行了：
Take care of myself in the better way (that可省略) you 【asked me to（承前省略了take care of myself）】.
或者：
Take care of myself in the better way (that) you 【wish/expect me to】
注意,用hope,可表达为：
Take care of myself in the better way (that) you hope.
因为,【hope sb to do sth】,这一结构是【错误】的!
其实,很久不用,你能写出这样的句子应该很不错的,只要有空再坚持学一点,你又会很出色的!
祝你开心如意!O(∩_∩)O~~