How to add a class to links inside posts text

Question

Is it possible? Sure.

You just need to re-declare the wpLink function that generates the link from the button.
For example like this

function wpse_181706_overwrite_wpLink() {
   if (wp_script_is('quicktags')){ ?>
    <script type="text/javascript">
            var wpLink;!function(e){var t,i,n,s,l={},a={},r="ontouchend"in document;wpLink={timeToTriggerRiver:150,minRiverAJAXDuration:200,riverBottomThreshold:5,keySensitivity:100,lastSearch:"",textarea:"",init:function(){l.wrap=e("#wp-link-wrap"),l.dialog=e("#wp-link"),l.backdrop=e("#wp-link-backdrop"),l.submit=e("#wp-link-submit"),l.close=e("#wp-link-close"),l.url=e("#url-field"),l.nonce=e("#_ajax_linking_nonce"),l.title=e("#link-title-field"),l.openInNewTab=e("#link-target-checkbox"),l.search=e("#search-field"),a.search=new n(e("#search-results")),a.recent=new n(e("#most-recent-results")),a.elements=l.dialog.find(".query-results"),l.queryNotice=e("#query-notice-message"),l.queryNoticeTextDefault=l.queryNotice.find(".query-notice-default"),l.queryNoticeTextHint=l.queryNotice.find(".query-notice-hint"),l.dialog.keydown(wpLink.keydown),l.dialog.keyup(wpLink.keyup),l.submit.click(function(e){e.preventDefault(),wpLink.update()}),l.close.add(l.backdrop).add("#wp-link-cancel a").click(function(e){e.preventDefault(),wpLink.close()}),e("#wp-link-search-toggle").on("click",wpLink.toggleInternalLinking),a.elements.on("river-select",wpLink.updateFields),l.search.on("focus.wplink",function(){l.queryNoticeTextDefault.hide(),l.queryNoticeTextHint.removeClass("screen-reader-text").show()}).on("blur.wplink",function(){l.queryNoticeTextDefault.show(),l.queryNoticeTextHint.addClass("screen-reader-text").hide()}),l.search.keyup(function(){var e=this;window.clearTimeout(i),i=window.setTimeout(function(){wpLink.searchInternalLinks.call(e)},500)})},open:function(i){var n;wpLink.range=null,i&&(window.wpActiveEditor=i),window.wpActiveEditor&&(this.textarea=e("#"+window.wpActiveEditor).get(0),"undefined"!=typeof tinymce&&(n=tinymce.get(wpActiveEditor),t=n&&!n.isHidden()?n:null,t&&tinymce.isIE&&(t.windowManager.bookmark=t.selection.getBookmark())),!wpLink.isMCE()&&document.selection&&(this.textarea.focus(),this.range=document.selection.createRange()),l.wrap.show(),l.backdrop.show(),wpLink.refresh(),e(document).trigger("wplink-open",l.wrap))},isMCE:function(){return t&&!t.isHidden()},refresh:function(){a.search.refresh(),a.recent.refresh(),wpLink.isMCE()?wpLink.mceRefresh():wpLink.setDefaultValues(),r?l.url.focus().blur():l.url.focus()[0].select(),a.recent.ul.children().length||a.recent.ajax()},mceRefresh:function(){var e;(e=t.dom.getParent(t.selection.getNode(),"A"))?(l.url.val(t.dom.getAttrib(e,"href")),l.title.val(t.dom.getAttrib(e,"title")),l.openInNewTab.prop("checked","_blank"===t.dom.getAttrib(e,"target")),l.submit.val(wpLinkL10n.update)):wpLink.setDefaultValues()},close:function(){wpLink.isMCE()?t.focus():(wpLink.textarea.focus(),wpLink.range&&(wpLink.range.moveToBookmark(wpLink.range.getBookmark()),wpLink.range.select())),l.backdrop.hide(),l.wrap.hide(),e(document).trigger("wplink-close",l.wrap)},getAttrs:function(){return{href:l.url.val(),title:l.title.val(),target:l.openInNewTab.prop("checked")?"_blank":"","class":"myclass"}},update:function(){wpLink.isMCE()?wpLink.mceUpdate():wpLink.htmlUpdate()},htmlUpdate:function(){var e,t,i,n,s,l,a,r=wpLink.textarea;r&&(e=wpLink.getAttrs(),e.href&&"http://"!=e.href&&(t="<a href=""+e.href+'"',e.title&&(l=e.title.replace(/</g,"&lt;").replace(/>/g,"&gt;").replace(/"/g,"&quot;"),t+=' title="'+l+'"'),e.target&&(t+=' target="'+e.target+'"'),e["class"]&&(t+=' class="myclass"'),t+=">",document.selection&&wpLink.range?(r.focus(),wpLink.range.text=t+wpLink.range.text+"</a>",wpLink.range.moveToBookmark(wpLink.range.getBookmark()),wpLink.range.select(),wpLink.range=null):"undefined"!=typeof r.selectionStart&&(i=r.selectionStart,n=r.selectionEnd,a=r.value.substring(i,n),t=t+a+"</a>",s=i+t.length,i==n&&(s-="</a>".length),r.value=r.value.substring(0,i)+t+r.value.substring(n,r.value.length),r.selectionStart=r.selectionEnd=s),wpLink.close(),r.focus()))},mceUpdate:function(){var e,i=wpLink.getAttrs();return wpLink.close(),t.focus(),tinymce.isIE&&t.selection.moveToBookmark(t.windowManager.bookmark),e=t.dom.getParent(t.selection.getNode(),"a[href]"),i.href&&"http://"!=i.href?(e?t.dom.setAttribs(e,i):t.execCommand("mceInsertLink",!1,i),void t.selection.collapse()):void t.execCommand("unlink")},updateFields:function(e,t){l.url.val(t.children(".item-permalink").val()),l.title.val(t.hasClass("no-title")?"":t.children(".item-title").text())},setDefaultValues:function(){var e=t&&t.selection.getContent(),i=/^[A-Z0-9._%+-]+@[A-Z0-9.-]+\.[A-Z]{2,4}$/i,n=/^(https?|ftp):\/\/[A-Z0-9.-]+\.[A-Z]{2,4}[^ "]*$/i;l.url.val(e&&i.test(e)?"mailto:"+e:e&&n.test(e)?e.replace(/&|&#0?38;/gi,"&"):"http://"),l.title.val(""),l.submit.val(wpLinkL10n.save)},searchInternalLinks:function(){var t,i=e(this),n=i.val();if(n.length>2){if(a.recent.hide(),a.search.show(),wpLink.lastSearch==n)return;wpLink.lastSearch=n,t=i.parent().find(".spinner").show(),a.search.change(n),a.search.ajax(function(){t.hide()})}else a.search.hide(),a.recent.show()},next:function(){a.search.next(),a.recent.next()},prev:function(){a.search.prev(),a.recent.prev()},keydown:function(t){var i,n,s=e.ui.keyCode;s.ESCAPE===t.keyCode?(wpLink.close(),t.stopImmediatePropagation()):s.TAB===t.keyCode&&(n=t.target.id,"wp-link-submit"!==n||t.shiftKey?"wp-link-close"===n&&t.shiftKey&&(l.submit.focus(),t.preventDefault()):(l.close.focus(),t.preventDefault())),(t.keyCode===s.UP||t.keyCode===s.DOWN)&&(!document.activeElement||"link-title-field"!==document.activeElement.id&&"url-field"!==document.activeElement.id)&&(i=t.keyCode===s.UP?"prev":"next",clearInterval(wpLink.keyInterval),wpLink[i](),wpLink.keyInterval=setInterval(wpLink[i],wpLink.keySensitivity),t.preventDefault())},keyup:function(t){var i=e.ui.keyCode;(t.which===i.UP||t.which===i.DOWN)&&(clearInterval(wpLink.keyInterval),t.preventDefault())},delayedCallback:function(e,t){var i,n,s,l;return t?(setTimeout(function(){return n?e.apply(l,s):void(i=!0)},t),function(){return i?e.apply(this,arguments):(s=arguments,l=this,void(n=!0))}):e},toggleInternalLinking:function(e){var t=l.wrap.hasClass("search-panel-visible");l.wrap.toggleClass("search-panel-visible",!t),setUserSetting("wplink",t?"0":"1"),l[t?"url":"search"].focus(),e.preventDefault()}},n=function(t,i){var n=this;this.element=t,this.ul=t.children("ul"),this.contentHeight=t.children("#link-selector-height"),this.waiting=t.find(".river-waiting"),this.change(i),this.refresh(),e("#wp-link .query-results, #wp-link #link-selector").scroll(function(){n.maybeLoad()}),t.on("click","li",function(t){n.select(e(this),t)})},e.extend(n.prototype,{refresh:function(){this.deselect(),this.visible=this.element.is(":visible")},show:function(){this.visible||(this.deselect(),this.element.show(),this.visible=!0)},hide:function(){this.element.hide(),this.visible=!1},select:function(e,t){var i,n,s,l;e.hasClass("unselectable")||e==this.selected||(this.deselect(),this.selected=e.addClass("selected"),i=e.outerHeight(),n=this.element.height(),s=e.position().top,l=this.element.scrollTop(),0>s?this.element.scrollTop(l+s):s+i>n&&this.element.scrollTop(l+s-n+i),this.element.trigger("river-select",[e,t,this]))},deselect:function(){this.selected&&this.selected.removeClass("selected"),this.selected=!1},prev:function(){if(this.visible){var e;this.selected&&(e=this.selected.prev("li"),e.length&&this.select(e))}},next:function(){if(this.visible){var t=this.selected?this.selected.next("li"):e("li:not(.unselectable):first",this.element);t.length&&this.select(t)}},ajax:function(e){var t=this,i=1==this.query.page?0:wpLink.minRiverAJAXDuration,n=wpLink.delayedCallback(function(i,n){t.process(i,n),e&&e(i,n)},i);this.query.ajax(n)},change:function(e){this.query&&this._search==e||(this._search=e,this.query=new s(e),this.element.scrollTop(0))},process:function(t,i){var n="",s=!0,l="",a=1==i.page;t?e.each(t,function(){l=s?"alternate":"",l+=this.title?"":" no-title",n+=l?'<li class="'+l+'">':"<li>",n+='<input type="hidden" class="item-permalink" value="'+this.permalink+'" />',n+='<span class="item-title">',n+=this.title?this.title:wpLinkL10n.noTitle,n+='</span><span class="item-info">'+this.info+"</span></li>",s=!s}):a&&(n+='<li class="unselectable no-matches-found"><span class="item-title"><em>'+wpLinkL10n.noMatchesFound+"</em></span></li>"),this.ul[a?"html":"append"](n)},maybeLoad:function(){var e=this,t=this.element,i=t.scrollTop()+t.height();!this.query.ready()||i<this.contentHeight.height()-wpLink.riverBottomThreshold||setTimeout(function(){var i=t.scrollTop(),n=i+t.height();!e.query.ready()||n<e.contentHeight.height()-wpLink.riverBottomThreshold||(e.waiting.show(),t.scrollTop(i+e.waiting.outerHeight()),e.ajax(function(){e.waiting.hide()}))},wpLink.timeToTriggerRiver)}}),s=function(e){this.page=1,this.allLoaded=!1,this.querying=!1,this.search=e},e.extend(s.prototype,{ready:function(){return!(this.querying||this.allLoaded)},ajax:function(t){var i=this,n={action:"wp-link-ajax",page:this.page,_ajax_linking_nonce:l.nonce.val()};this.search&&(n.search=this.search),this.querying=!0,e.post(ajaxurl,n,function(e){i.page++,i.querying=!1,i.allLoaded=!e,t(e,n)},"json")}}),e(document).ready(wpLink.init)}(jQuery);
    </script>
    <?php }
}
add_action( 'admin_print_footer_scripts', 'wpse_181706_overwrite_wpLink' );

Put that in your functions.php and search and replace “myclass” with what you want.
(edit: javascript borrowed and modified from Rel Nofollow Checkbox -plugin)

Should you do it this way (in my opinion)? Probably not.

Either do like Laurence said and just use CSS. If you really need classes I would probably use javascript/jQuery to add the classes on page load, something like this

jQuery(document).ready(function($) {
    $('article a').addClass("myclass");
});