Putting your hair through more than one chemical process is very damaging. It is particularly damaging to dye your hair a darker color, then bleach streaks. In general, you should never do two processes at home. If you dye your own hair then apply highlights, it can cause any number of weird chemical reactions to occur. The streaks may come out an unintended color. Your hair may be highly damaged. It may even break off and fall out.
If you want to do multiple processes, the best thing to do is go to a salon. A good colorist will know which dyes can be combined without causing much damage to your hair. The colorist may suggest a waiting period between two processes or special conditioning products that can help reduce the damage to your hair. Over-processed hair needs to be trimmed frequently to help eliminate split ends and breakage. Have both processes done with the same stylist and make an appointment for your next trim while you're there.
There are safe ways to change your hair color and add highlights at home. New products, like Revlon Custom Effects, are a kit in one box to change your hair color and add highlights at the same time. There are also two-step dye and highlight products like Couleur Experte by L'Oreal. It is designed to give hair a perfectly matched, multi-tonal look without over-processing or damaging your hair.