Intenta hundirme con cada comentario que hace sobre mi persona

Intenta hundirme con cada comentario que hace sobre mi persona
 

I try it first:
He tries to sink me with every comment he makes about my person.
Is it correct?

No. A mi me suena torpe.
A better way of saying this is: He tries to put me down with every comment he makes about me.
Instead of put me down you can say (in New York at least) throw me under the bus.
There are other ways too--betray for example.

"to bad mouth" for example.

thank you

Sure - "Put-down" type people - first "putting one across your bow" (firing
a warning shot), then "Putting you down" (degrading, discrediting), and then "one-upping" you...and maybe sinking you, maybe not...it's a game, right?