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If you want to use essential oils for eczema, always spot test first to make sure you don't react badly to either the essential oil or the carrier oil. Using authentic, high-quality essential oils is a must, as low-quality oils are likely to do more harm than good.
Start by testing carrier oils without any added essential oil. If you don't have an adverse reaction to the carrier oil within 24 hours, then make a natural treatment using one of the following best essential oils for eczema relief:

If you are new to essential oils, start with a one percent dilution, that is, 1 drop of essential oil in 1 teaspoon of carrier oil. Test the blend on a small area before using it on a larger area.
Essential Oil Compress for Eczema
Source: Massage & Aromatherapy: Simple Techniques to Use at Home:
Symptom Relief Cream
Mix the essential oils into the vitamin E cream. Apply to skin as needed.
Source: The Healing Intelligence of Essential Oils
Eczema Relief Oil
Blend the following in a small dark bottle:
Source: The Complete Book of Essential Oils and Aromatherapy
Eczema Relief Bath
Makes 5 to 6 baths.
Source: Essential Oils for Healing
Compress For Weeping Eczema
Source: Healing With Aromatherapy
Weeping Eczema Aromatherapy Cream
To use, apply a small amount of the cream to affected areas.
Source: The Complete Aromatherapy & Essential Oils Handbook for Everyday Wellness
Aromatherapy Blend For Dry Eczema
Mix thoroughly in a small bottle. Shake before use. Apply the blend to lesions 4 times a day. (If the aloe vera gel is from a fresh plant, keep this blend refrigerated.)
Source: Healing With Aromatherapy
Dry Eczema Relief
Drop
essential oils into a dark bottle and then add the carrier oils. Shake
well. Let set in a cool, dry place for 24 hours before using.
To
use: Clean the affected area using one part baking soda to eight parts
water. Pat dry. Shake the essential oil blend. Gently massage a small
amount of oil into the affected area. Repeat two or three times a day.
Source: Hands-On Healing Remedies

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Buy essential oils at Aromatics International or Rocky Mountain Essential Oils.
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Photo Credit: Jambula [Public domain] via Wikimedia Commons