What layer of skin contains the blood vessels and nerves?

Dermis

The skin has two layers : Epidermis, the epithelial layer and Dermis , the connective tissue layer. This two layer rest on another connective tissue layer called Hypodermis or subcutaneous tissue.

The connective tissue layer of the skin (dermis) contains a rich network of blood and lymphatic vessels. Epidermis does not have blood vessels.

The following diagram shows the layer of the skin, see all the blood vessels are in dermis.

Most of the nerve tissue supplying the skin, including encapsulated and expanded receptors, are in the dermis. But epidermis also contains some nerve tissue (the free nerve endings). Skin is the most extensive sensory receptor of the body, and both the two layers of it contain nerve tissue.

Following diagram is showing the nerve tissues in the skin, here we can see the free nerve endings are in the epidermis, and all other nerve endings are in the dermis :

So, the dermis contains all the blood vessels and most of the nerve tissue of the skin.