How do I know my Eevee is ready to evolve into Umbreon?

There's a friendship checker in Goldenrod City, in a house near the cycle shop. The ratings that she will say are:

• 250-255 - "It looks really happy! It must love you a lot."
• 200-249 - "I get the feeling that it really trusts you."
• 150-199 - "It's friendly toward you. It looks sort of happy."
• 100-149 - "It's quite cute."
• 50-99 - "You should treat it better. It's not used to you."
• 0-49 - "It doesn't seem to like you at all. It looks mean."

According to this source, the exact amount of friendship you'll need for Eevee to evolve into Umbreon is 220. Just remember to level it up at night time once you reach this amount of friendship!

Now I can't find an exact source of all the walking-pokemon interactions and what they represent in terms of happiness, although from memory once the speech bubbles start to show love hearts, it means it's generally high friendship. I think when your Pokemon dances for joy or hugs you, that represents the highest tier of friendship.

You can read more about interacting with the walking Pokemon here