Recently purchased a Nikon D7000 with a pop up flash. I purchased a small plastic flash diffuser ( attaches to the hot shoe top mount )and it still seems to produce harsher light for portraits than I want.
Light that comes from the camera's position isn't going to give you nice light for portraits. Portraits are all about directional lighting that can create highlights and shadows on your subject's face. So yes, off-camera lighting (preferably flash) is the way to go.
Here is the link to 'Lighting 101' on the Strobist blog mentioned by Derrel.
Strobist: Lighting 101
Here is a link to a site that sells 'strobist kits'. You certainly don't have to buy from them, but take note of the equipment in the kit.
Midwest Photo Exchange Strobist® Kits - STROBIST®
As for trying to mimic natural light...that can be easy, or very tricky...depending on the situation. Typical nice 'natural' light is soft (not counting direct sunlight). For light to be soft, the light source has to be large in relation to the subject. So a large window, or the open sky (while standing in the shade)...these are large light sources which produce soft light. In a studio, will will use something like an umbrella, a softbox or a large diffusion panel.
An off camera flash, by itself will produce hard light because it's so small. Using an umbrella or softbox can really help...but it takes a really large umbrella or softbox to emulate that really soft light that you want. And a small flash unit probably doesn't have enough power to use a large modifier like that (you'd need a studio light). So, when you want the portability of a flash unit, you have to compromise on the power and/or softness of the light you can create.
Although, bouncing the light off of a wall or ceiling etc. can give you nice soft directional light, without any modifier...by you are reliant on a good surface to bounce off of.
Another option would be to buy/build a diffusion panel. You can then shoot the flash though the panel for soft light...or you could hold the panel up between your subject and the sun, which makes for beautiful soft light. Of course, this requires an assistant to hold the panel.
Another issue is knowing how to properly use flash and how to control the ratio between flash and ambient light. If you want to (and can achive) a nice balance between the flash and the ambient light, then you probably don't need to use really soft light.
I use something like this...
Amazon.com: DMKFoto Softbox for Speedlight and Flash with Stand: Camera & Photo