First make sure you can reach your server - you app may be running locally but it doesn't use the same server as the one from your dockerfile e.g. npm start starts a local server.
Remove everything related to the react app stage from your dockerfile and just make sure you can hit your server on port 5000 and that it servers pages - you can host an index.html with <body>Hello world</body> or something. When you're able to do that it's just a matter of adding the react app bundled files to the server static (public) folder
Is your node server listening on port 5000 ?
Typically with node js server you have to set a PORT and HOST env variables that are used by the server like process.env.PORT and process.env.HOST
Other issues
# start app
CMD ["npm", "start"]
npm start is (usually) used to start a local server while you develop your react app
Instead of starting the app you should run a bundle command that will produce static files to be hosted - js html etc...
Typically npm run build
This should make a folder ./dist or ./build with all the static content you should put on the server (perhaps that's your /client folder for?)
Some tweaks
# pull official base image
FROM node:13.12.0-alpine AS builder
# set working directory
WORKDIR /client
# Copy source
COPY . .
# add `/app/node_modules/.bin` to $PATH (you probably don't need this)
ENV PATH /client/node_modules/.bin:$PATH
# install app dependencies
RUN npm install --silent
RUN npm install [email protected] -g --silent (you can move this to dependencies)
# Build bundle
RUN npm run build
# Next stage
FROM node:12
# Create app directory
WORKDIR ./
# Copy source
COPY . .
RUN npm install
# If you are building your code for production
# RUN npm ci --only=production
# Copy js bundle
COPY --from=builder /client/dist ./public
EXPOSE 5000
CMD [ "node", "index.js" ]
This might need a little more tweaking but that's the general idea
you don't need to install react-scripts globally it can just be a local dependency - it's used by npm start and npm build to build your app so you depend on it - it should be part of package.json dependencies
Adding /client/node_modules/.bin to PATH shouldn't be needed, maybe it's a leftover or a tell that something else isn't properly setup