For my comic's logo, I made it totally from scratch. I did a rough sketch of it on paper first, then used the dreaded illustrator pen tool to turn it into a vector. The "shenanigans" part is the font I use for my dialogue, which I created from my own handwriting using this site.
How you go about designing your logo can vary greatly depending on your comic's title. If it's more than one word long, are the words on different lines? Is one word bigger and more emphasized? Maybe you'll want more than one version of your logo, such as one that has the full title on a single line for your banner, and one with the words stacked for book covers and other vertical layouts. Making sure your logo can fit wherever you plan on sticking it is important!
Although, if you are considering hiring someone to create your logo for you, designing comic logos is something I've thought about. So feel free to talk with me about that if you're interested!