1)
Why does it mention Yisrael?
1.
Radak: They must thank Hashem for David's kingship. David conquered their enemies even in the days of Sha'ul, and all the more so [he will] now that he is king, they need not fear enemies!
2.
Malbim: They must thank Hashem more than other nations, for they received more good. Chovos ha'Levavos says that this is why Yisrael are obligated in 613 Mitzvos, and Kohanim received 24 gifts corresponding to 24 Mitzvos of Kehunah.
2)
Why does it repeat "Ki l'Olam Chasdo"?
1.
Radak: They were slaves in their Galus, and now they are masters.
