The CLASSPATH environment variable is only used by the java.exe command and even then only when the command is invoked without any of the -cp, -classpath, -jar arguments. The CLASSPATH environment variable is ignored by IDEs like Eclipse, Netbeans and IDEA. See also java.lang.ClassNotFoundException in spite of using CLASSPATH environment variable.
The Build Path is only for libraries which are required to get the project’s code to compile. Manually placing JAR in /WEB-INF/lib, or setting the Deployment Assembly, or letting an external build system like Maven place the <dependency> as JAR in /WEB-INF/lib of produced WAR during the build, is only for libraries which are required to get the code to deploy and run on the target environment too. Do note that you’re not supposed to create subfolders in /WEB-INF/lib. The JARs have to be placed in the root.
Some libraries are already provided by the target JEE server or servletcontainer, such as JSP, Servlet, EL, etc. So you do not need put JARs of those libraries in /WEB-INF/lib. Moreover, it would only cause classloading trouble. It’s sufficient to (indirectly) specify them in Build Path only. In Eclipse, you normally do that by setting the Targeted Runtime accordingly. It will automatically end up in Build Path. You do not need to manually add them to Build Path. See also How do I import the javax.servlet / jakarta.servlet API in my Eclipse project?
Other libraries, usually 3rd party ones like Apache Commons, JDBC drivers and JEE libraries which are not provided by the target servletcontainer (e.g. Tomcat doesn’t support many JEE libraries out the box such as JSF, JSTL, CDI, JPA, EJB, etc), need to end up in /WEB-INF/lib. You can just copy and paste the physical JAR files in there. You do not necessarily need to specify it in Build Path. Only perhaps when you already have it as User Library, but you should then use Deployment assembly setting for this instead. See also ClassNotFoundException when using User Libraries in Eclipse build path.
In case you’re using Maven, then you need to make absolutely sure that you mark libraries as <scope>provided</scope> if those are already provided by the target runtime, such as JEE, Servlet, EL, etc in case you deploy to WildFly, TomEE, etc. This way they won’t end up in /WEB-INF/lib of produced WAR (and potentially cause conflicts with server-bundled libraries), but they will end up in Eclipse’s Build Path (and get the project’s code to compile). See also How to properly install and configure JSF libraries via Maven?