[ε] in the French language sounds like a double vowel, close to the English [ei], how should I pronounce it correctly? Besides, a French [ε] sound sounds obviously different from an English [ε]. Can you help me with my question?
[ɛ] is not a diphthong (what you call a "double vowel") it is a mid-open front vowel. On the wiktionary you can listen to the word tête and have the IPA phonetic transcription besides. Moreover for this word you can compare the pronunciation in different types of French, you can see that the word is pronounced with the sound [ɛ] in France and with a diphthong [æi] in Quebec.
For your information, an English word with the sound [ɛ]: dress.
Here are two samples of the French [ɛ]:
As an example, consider the letter L which sounds roughly the same ([ɛl]) in French and in English.