Unlike shortening a dining chair, raising it is a rather difficult task in terms of carpentry. It's difficult because the spacers will have to be exactly the same height, be able to support the weight, be safely attached and match the design of the chair legs. All of this combined means only experienced carpenters are likely to be able to pull off a good enough job to make this a viable option. There is a solution on the market which makes this task possible at home.
Work out how much you want to raise the chair. The best way to do this is to sit down and estimate where a comfortable position would be.
Decide what kind of raisers you want. If the chair needs to be raised a few millimetres then felt stoppers glued to the chair leg bottoms will be fine, if you need a few centimetres then plastic raisers will be needed.
Purchase the appropriate raisers. These are widely available and come in a variety of styles to suit your dining room set.
Fit the raisers to the bottom of all four legs. The plastic versions fit on over the leg itself, the felt stoppers are self adhesive.
If the chair needs to be raised a lot, there is little that can be done as adding extra bits onto the chair legs will weaken the overall design.