There are various ways you can do this:
Method 1: Snapshoting
You can go through the container filesystem:
# find ID of your running container:
docker ps
# create image (snapshot) from container filesystem
docker commit 12345678904b5 mysnapshot
# explore this filesystem using bash (for example)
docker run -t -i mysnapshot /bin/bash
By doing this you can check the filesystem of running container in that moment in time. The container would still be in running condition with no future changes included.
After, you could delete this snapshot:
docker rmi mysnapshot
Method 2: SSH
If you want something continuous, you can use SSHD. You'll have to install it on your container and then execute SSHD Daemon:
docker run -d -p 22 mysnapshot /usr/sbin/sshd -D
# you need to find out which port to connect:
docker ps
This way you'll be able to run you application using SSH.
Method 3: docker exec
If bash is present on your container you can directly use shell on your container by using:
docker exec -t -i mycontainer /bin/bash
Check Docker CLI documentation