This first question gives me the opportunity to address the complexities of understanding the Bible based on the original languages in which it was written. Every word has both a definition and a meaning. The definition of a word is the one word or phrase that you might find in a lexicon or dictionary. The meaning of a word is an understanding of the word in association with the other words or phrases that are in association with that word.
In this case, a good working definition of chayil is strength, power, dominance, or even wealth. The meaning of this word, as well as which definition applies here, is determined by context. The context is describing the chayil woman as possessing certain attributes of her character. So, a good meaning for this word would be "capacities of strength." We arrive at this meaning because after introducing the chayil woman, the writer of the proverb then lists 13 characteristics of her strength or capacity.
Proverbs 31:10 [1] says, "A virtuous woman who can find..." This tells us that a woman of virtue, or a woman of strength of capacity, is a rare find. In fact it indicates that she must be found. The message of this proverb is important today, especially with the worldly emphasis on outward beauty and material gain. This proverb is making emphasis on the inward beauty or inward capacities of the woman, which is also emphasized in 1 Peter 3:3-4 [2]. Certainly the verses at the end of the proverb express it all, "...elegance is deceptive and beauty is as a breath, but a woman who fears Yahweh, she herself shall be praised. Give her the fruit of her hands, and may her works praise her in the gates." Proverbs 31:20-31 [3] (Literal translation)