In every blog site there is one Comments list. Always called "Comments". In this list there is a column called Post ID.
Therefore if you already have your post, you grab the ID, look at the comments list and query for all comments where Post ID = Blog Post ID. A simple CSOM CAML query should do it. (Looking at the Post Id column it is stored as 1;#1 so you might have to do some manipulation before performing the query)