To enable Tomcat 8 as a CDI and JAX-RS container you need to put all dependencies of these artifact in the tomcat/lib directory:

<dependency>
    <groupId>org.jboss.weld.servlet</groupId>
    <artifactId>weld-servlet</artifactId>
    <version>2.3.3.Final</version>
  </dependency>
  <dependency>
    <groupId>org.glassfish.jersey.ext.cdi</groupId>
    <artifactId>jersey-cdi1x-servlet</artifactId>
    <version>2.22.2</version>
  </dependency>
  <dependency>
    <groupId>org.glassfish.jersey.containers</groupId>
    <artifactId>jersey-container-servlet</artifactId>
    <version>2.22.2</version>
  </dependency>

All above dependencies can be build with this small Maven project: Tomcat-JAXRS-CDI

Just run

mvn package

and unzip the zip file in the target directory to the tomcat/lib directory.