Dynamic List Application as first Vue.js App
At the moment all your current hobbies only consists of a name. I think you could use a simple string instead of wrapping it in an object (for point 8 this is not possible though, of course).
No need to check if (!this.deleted) before setting this.deleted = true;
setStyle should be a computed property
You can use @click instead of v-on:click, you can also use :style instead of v-bind:style (just syntactic sugar)
I would put the v-if="deleted" on the &lt;div id="deleted-message"&gt; instead of the inner &lt;p&gt; element.