Step 1 : Login into tumblr 
Step 2 : Click Account icon [] >> Settings

Step 3 : From right handside , select the blog you wish to delete * 
Step 4 : Scroll end of the blog setting , you would find Delete blog button 

Step 5 : Fill the login information to delete the blog

Step 6 : confirmation message will show up 
* Note : if you will delete your primary blog , you most delete your account , which will be similar steps , just the button would be rather than delete blog , will be delete account