I hate doing the dishes and never have anyone over because it gets to levels no human should ever see. Can or should I get a countertop dishwasher? Will my landlord hate me? What if I don't have any countertop for it?
You could totally get a countertop dishwasher. There are even non-countertop versions, too. For instance, The Sweethome recommends this portable dishwasher on wheels, for $450.
In the strictly-countertop world, though, people on Amazon seem to be pretty happy with this top-rated $230 model. I also like this bit from its highest-rated comment, which reminds me of your question: "My dislike of doing dishes turned most of the counter space of my small Manhattan kitchen into physics-defying stacks of procrastination. This countertop dishwasher takes up about 60% of usable counter space in my apartment, a sacrifice I'm willing to make."
And as far as the landlord situation goes, unless it's explicitly prohibited in your lease, it's perfectly legal to have a countertop dishwasher. And it won't affect the utilities, either: A countertop dishwasher cycle apparently uses less hot water than a manual wash ("Each wash cycle, which holds 4-6 place settings, uses about 3 gallons of water").
I say go for it.