Psalm 136 uses a repeating refrain, not just the noted two verses:
כִּי לְעוֹלָם חַסְדּוֹ
The last word, chasdo (3pm possesive, from chesed), is generally translated as "kindness" or "loving kindness". The word "grace" is a different word, chein. See, for example, Psalm 145:8:
חַנּוּן וְרַחוּם יְהוָה
The LORD is gracious, and full of compassion
"Gracious" here is the first Hebrew word (from chein) and "full of compassion" (sometimes "merciful") is the second.
Focusing on chesed, which seems to be the core of the question, the Strongs entry says (down at the end):
(2) in a bad sense, zeal, ardour against anyone, envy (hence reproach)
It gives gives two text references for this meaning, Lev 20:17 and Prov 14:34 (and "some would say Job 6:14").
Lev 20:17 is:
... וְאִישׁ אֲשֶׁר-יִקַּח אֶת-אֲחֹתוֹ בַּת-אָבִיו אוֹ בַת-אִמּוֹ וְרָאָה אֶת-עֶרְוָתָהּ וְהִיא-תִרְאֶה אֶת-עֶרְוָתוֹ, חֶסֶד הוּא
And if a man shall take his sister, his father's daughter, or his mother's daughter, and see her nakedness, and she see his nakedness: it is a shameful thing...
"Chesed" here is translated "shameful thing". The translation used on the Rashi site (from Judaica Press) uses "disgraceful act". Rashi, a medieval scholar and compiler, connects this act of marrying one's sister to an earlier one:
it is a disgraceful act: The Aramaic term for “disgrace” is חִסוּדָא. - [see Onkelos on Gen. 34:14] Its Midrashic interpretation, however, is: If you [object and] say, “But Cain married his sister!” [the answer is:] the Omnipresent [in permitting this marriage,] performed an act of kindness (חֶסֶד), to build His world through him, as it is said: “the world is built on kindness (חֶסֶד) ” (Ps. 89:3). - [Torath Kohanim 20:116]
Marrying one's sister is a shameful act but in the case of Cain it was necessary, so Rashi understands that to be a divine kindness. (Why this doesn't apply to Seth is not explained here.) In other words, he sees the text here, which would normally be understood as "kindness" but talks about marrying a sister, as indicating that marrying one's sister could be a kindness, but not here in Leviticus. (I am probably missing some additional connections here, but this is what I've got.)
Now, turning to Proverbs, the passage in question is:
צְדָקָה תְרוֹמֵם גּוֹי וְחֶסֶד לְאֻמִּים חַטָּאת:
Righteousness exalteth a nation; but sin is a reproach to any people. (JPS)
Charity will elevate a nation, but the kindness of the kingdoms is sin. (Judaic Press)
The latter is closer to the dominant meaning of chesed. Why would JPS and others instead render "reproach"? Rashi offers this interpretation:
but the kindness of the kingdoms is sin: They are the heathens, who rob one to give another.
In other words, it appears to be a kindness, but it's a false kindness -- giving to another is kind, but robbing to get the money to give is certainly not. Rashi does not cite a source for this interpration.
As for Job 6:14, the passage is talking about withholding kindness, but I don't see why "some would say" that's relevant here.
"Kindness" is by far the dominant meaning for chesed. There are some special cases that are negative but can arguably be made to fit with the "kindness" theme nonetheless.
Please note: This answer was written for a neutral, academic audience
and is not intended to be interpreted in the context of a religious
belief or doctrine.