Absolutely. The simplest mode (multicolor) has this great restriction: For every 8x8 cell (4x8, if you look at them like widepixels), you can only have 3 colors. Plus, you get one color that you can use everywhere, which must be the same for the whole picture. So it's 4 max colors per cell.
Don't be afraid of this limitation. You can easily apply it for your picture with minimal changes, and often the changes that it'll force you to make will open your mind to new shapes of clusters and smart solutions to problems. Most of the time we use too many colors. The c64 can teach a pixel artist that they can do more with less in a lot of cases.
Also a handy thing: if you're working in a way that might result to stray square pixels in such an image, make a habit of compressing the 320 width of the image to 160, then back to 320. Do it before the final image is produced as well. That'll take care of any stray square pixels.
As to your image as it stands right now, I really like it. There are many little bits I would change, but with an entry into the c64 I would suggest first to make a few more pictures, and to do them in multicolor or hi-res and then work on the nitpicks.