See http://www.w3schools.com/cssref/sel_after.asp
The selector for this would be
h1.posttitle:after {} or .posttitle h1:after {} (Without trying, I’m not sure which is correct.)
Sorry. I may have mis-lead you about what I was looking for. Rather than controlling the style, I want to incorporate the ::after selector along with any h1.posttitle I manually create in my html. How do I use the ::after selector in my html?
Here’s a solution that gives the result I want, but it’s not utilizing the ::after style already set by my theme:

Thanks.
You don’t need to incorporate it into your HTML. If you use an :after selector in your CSS, the content of that selector will appear.
So, here’s how my theme styles post titles:

And here’s my solution to manually style any text in the same way:
<header class="post-header fix"><h1 class="posttitle">ABOUT US</h1></header>
Seem correct?
I still don’t understand how the theme inserts ::after in the html or why I don’t need to insert it in my html. It appears to be a clearfix.
You do not put ::after in your html. Period.
Ah! It’s in the theme’s css! Sorry! I get it now.
.fix::after {
content: ".";
display: block;
height: 0;
clear: both;
visibility: hidden;
}
I was confused by why it showed up in my Inspector.
The inspector does not necessarily show you the exact HTML.