1. What you HAVE is a VECTOR image. AI is a vector program. It doesn't deal with DPI at all. It only deals with math. A "vector". So you can scale up and down infinitely. All the points are plotted mathmatically. NO dpi.
2. What you NEED is a RASTER image... a bitmap, which is based on a finite amount of data contained in a a finite amount of space. Programs like Photoshop deal with raster images (used for photo correction, manipulation & in your case, output to gifs & jpgs) One piece of information in one area of space... DPI, dots per square inch (or Pixels per square inch on a monitor) Monitors only have 72 DOTS (or pixels) per inch available. If you are viewing a high-res bitmap, you can zoom in and see the tiny details. You are still only seeing 72 DPI. Same with AI, when you zoom in, you are still only viewing 72 DPI. You might be able to PRINT at a higher resolution, but all you can ever see on a monitor is roughly 72 DPI at any one time.
3. If I had your source file. I could make it look roughly the same as your ai file at the same size, this includes quality & color uniformity. Starting with your flawed screengrab, though,.. doesn't get us very far.
This is just a summary.... Image production is a professional field, and there are a lot of issues regarding image quality on the web, and in print. To be able to take a basic illustrator file, and make it an acceptable JPG requires a little research, experience, & practice.
Give me your file, and I'll show you how it's done.