Before I give this feedback away, i’d like to point out that every person is different and you should be flexible on doing on what works or does not work for you.

So before we move forward, id like to share this link.

This guy is really good at explaining the difference on being an illustrator to a designer.

So, things to consider when being an emerging artist:

• What came first, the egg or the chicken? In this case scenario how ever, its the portfolio and the website. A great portfolio might work very well as a hard copy brochure or a PDF, but nothing works better than giving a link (preferably just your name, or a incredibly short and/or easy name to remember) to show your work. So you should look for options on working on your website. But I do not think that you should necessarily work in just one, or the other. But having a basic, simple, minimalist, and straight to the point website works the best. Does not matter how ugly is your website if you have an amazing portfolio to show. You can always make your website better, but your website is simply and merely a tool to show your work and a frame of your work.
  I find this a very good example. lazaruz
• Your portfolio should sell well what was the main objective/brief of the drawing. What are you trying to sell? what are you communicating? what was the purpose of your drawing? Who is the creature/character? and what the history and story behind it? But most importantly: What makes great character design great? A thoughtful process and a wow factor. When you create a character/creature. Ask your self as many questions as you can continually on what makes that character the character and its hierarchy to follow. Its easy to stop to think what or who are you drawing, and just texture, color and paint away. But sometimes you have to pause, look back, and ask your self the questions of what makes it.
  I found this pinterest board, and its stuff its delicious: http://www.pinterest.com/trabello/character-concept-art/ Comparing the works in between can be a good example.
• Presentation is the last key point on showcasing your work, is your work good enough to show? Avoid your art to be daft and sketchy qualities. (unless you indicate well that its a WIP or sketch) try to make sure that the brush strokes are not obviously seen unless its the nature of your art piece. Be careful and be sure to deliver the last 5% of your work. Also, framing, angle and use of white space is important. If you are doing to be a character designer, you have to show graphic design qualities that are shown in your work. For the nature of design, its important that the whole body is displayed, preferably in front, back and 3/4. If you have a design that might display different color and/or variations, that differs. In other case scenarios you might show the subject in different lighting conditions. Some characters could show different stages such as damage, age, for different angles of the different parts to clearly show better its intention from the design.
• Last, some characters work really well on having a white background, but also try mid range greys. The reason behind this, its because its much easier to distinguish the different values. Also, the more research, the better. Do not be content with keep working on the first iteration, if anything, work on creating a lot of concepts, silhouettes and forms before picking one. And never be attached on anything you do, because changes are, when you are working, your boss/supervisor might think your design is amazing or great, but the design might not work for its purpose, and then it might be have to be done all over again.

• Network. Find how you can meet people that are in the same industry and have the same interests. Ask around and see who you can meet. And learn to work in production environments. Time your self how long it is taking you to do the different designs and think how long can you take to do the different areas.

1, make a website, do not worry on how good/bad it is, but on having your best work on site

2, before on starting new pieces or works, think well or ask someone to give you a brief for practice

3, presentation, framing, last qualities, placing your work in a good frame with the cherry on the top

4, detachment to your work and experimentation

What I would recommend for new aspiring artists or designers to do to build a new portfolio, its to focus on a project where several animals/creatures/people might be in an set environment. Does not mind if its a fantasy, sci fi or earthly place. But do several designs in the same universe to give a chance for the viewer to be immersed in a world where its all plausible. Before you start drawing of course, spend good thoughtful time on things as mood, setting, color, feel, style, etc.

To do this, make a word/google doc that explains the different qualities of the story/place in 1 to 2 pages. You can share this with a friend for feedback.

Even better, how about coming up with a well constructed character template?

This might be a little bit too in depth, but you get the idea!
http://fyuvix.deviantart.com/art/Character-Profile-Form-36823983