The interdependence between the dharma (virtue), artha (wealth) and kama (pleasure) can be understood well form the words of Bhimasena in Mahabharata. Note that the words of Chanakya are mainly intended for a Kshatriya and hence my answer is.
I will divide my answer into two parts. First part addresses the dependence of dharma on artha and the second part addresses the dependence of kama on artha
#1: Dependence of dharma on artha:
Artha is a requisite for Dharma. Dharma exists in multiple forms. In general, (for Kshetriya), sacrifices, respecting guests, charity etc., are the highest virtues and are possible to do only by wealth. Thus several kinds of dharma cannot be performed without artha.
Gift, sacrifice, respect for the wise, study of the Vedas, and honesty, these, O king, constitute the highest virtue and are
efficacious both here and hereafter. These virtues, however, cannot be
attained by one that hath no wealth, even if, O tiger among men, he
may have infinite other accomplishments. The whole universe, O king,
dependeth upon virtue. There is nothing higher than virtue. And
virtue, O king, is attainable by one that hath plenty of wealth.
Wealth cannot be earned by leading a mendicant life, nor by a life of
feebleness. Wealth, however, can be earned by intelligence directed by
virtue. In thy case, O king, begging, which is successful with
Brahmanas, hath been forbidden. Therefore, O bull amongst men, strive
for the acquisition of wealth by exerting thy might and energy.
Neither mendicancy, nor the life of a Sudra is what is proper for
thee. Might and energy constitute the virtue of the Kshatriya in
#2: Dependence of Kama on artha:
Kama can be acquired by artha, as well as dharma. (Internal) Pleasure can be obtained by the person who performs dharma, which needs wealth (#1) . (External) Pleasure can be obtained from the five senses using the objects of enjoyment, which also needs wealth to gain them.
Destitute of virtue and wealth such a man, indulging in pleasure at
will, at the expiration of his period of indulgence, meeteth with
certain death, like a fish when the water in which it liveth hath been
dried up. It is for these reasons that they that are wise are ever
careful of both virtue and wealth, for a union of virtue and wealth is
the essential requisite of pleasure, as fuel is the essential
requisite of fire. Pleasure hath always virtue for its root, and
virtue also is united with pleasure. Know, O monarch, that both are
dependent on each other like the ocean and the clouds, the ocean
causing the clouds and the clouds filling the ocean. The joy that one
feeleth in consequence of contact with objects of touch or of
possession of wealth, is what is called pleasure. It existeth in the
mind, having no corporeal existence that one can see. ........ And, O king, as a fowler killeth the birds we see, so doth sin
slay the creatures of the world. He, therefore, who misled by pleasure
or covetousness, beholdeth not the nature of virtue, deserveth to be
slain by all, and becometh wretched both here and here-after. It is
evident, O king, that thou knowest that pleasure may be derived from
the possession of various objects of enjoyment. Thou also well knowest
their ordinary states, as well as the great changes they undergo. At
their loss or disappearance occasioned by decrepitude or death,
ariseth what is called distress. That distress, O king, hath now
overtaken us. The joy that ariseth from the five senses, the intellect
and the heart, being directed to the objects proper to each, is called
pleasure. That pleasure, O king, is, as I think, one of the best
fruits of our actions.
In fact, the three are interdependent for Kshetriyas.
Thus, O monarch, one should regard virtue, wealth and pleasure one
after another. One should not devote one self to virtue alone, nor
regard wealth as the highest object of one's wishes, nor pleasure, but
should ever pursue all three. The scriptures ordain that one should
seek virtue in the morning, wealth at noon, and pleasure in the
evening. The scriptures also ordain that one should seek pleasure in
the first portion of life, wealth in the second, and virtue in the
last. And, O thou foremost of speakers, they that are wise and fully
conversant with proper division of time, pursue all three, virtue,
wealth, and pleasure, dividing their time duly. O son of the Kuru
race, whether independence of these (three), or their possession is
the better for those that desire happiness, should be settled by thee
after careful thought. And thou shouldst then, O king, unhesitatingly
act either for acquiring them, or abandoning them all. For he who
liveth wavering between the two doubtingly, leadeth a wretched life.
It is well known that thy behaviour is ever regulated by virtue.
Knowing this thy friends counsel thee to act.
All the words of Bhimasena are taken from the [section 38, Arjunabhigamana Parva, Vana Parva, The Mahabharata]