Examine the tree carefully to determine which branches need to be cut back. You can also remove some of the lower branches if necessary to get access to the rest of the branches.
Cut off any dead or diseased branches at the very base of the branch against the tree. Be careful when cutting that you do not chip or gouge at the remaining bark. Wipe the blades of your pruning tool with rubbing alcohol between cuts to prevent spreading disease.
Cut out any healthy branches that need to be removed in order to thin out specific areas around the dogwood tree, using clean pruning shears. Again, make your cut at the very base of the branch and ensure you do not injure the bark as you cut. Dogwood bark is very thin and not especially resistant to damage.
Continue making your cuts until you have removed all dead or diseased branches and the branches have been sufficiently thinned out, or until you have achieved the effect you are going for.