The first example of Vasudeva is in Bhagavad-Gita,
chapter 7, verse 19 below:

bahunam janmanam ante jnanavam mam prapadyate
vasudevah sarvam iti sa mahatma su-durlabah

Translation
Thus after innumerable births one　 who is perfected in wisdom,
understanding completely that Vasudeva is the ultimate cause of
 all causes surrenders unto Me; such a great soul is very rare.

In Srimad Bhagavad-Gita, chapter 10, verse 37,
Lord Krishna declares again:

vrsinam vasudeva'smi pandavanam dhananjayah
muninam apy aham vyasah kavinam usana kavih

Translation
Of the descendants of Vrisni I am Vasudeva,
of the Pandavas I am Dhananjaya,
of the holy sages I am Vyasadeva and
among great scholars of scriptures I am Usana.

Vedavyasa also refers to Lord Krishna as Vasudeva when
describing the phenomenal universal form Lord Krishna
displayed in Srimad Bhagavad-Gita, chapter 11, verse 50:

ity arjunam vasudevas tathoktva svakam rupam darsayamasa bhuyah
asvasayamasa ca bhitam enam bhutva punah saumya-vapur mahatma

Translation
Vasudeva speaking to Arjuna in this way,
exhibited His four-armed form and then again appeared in
His beautiful two armed form as the Supreme Lord Krishna and
pacified Arjuna's fears.

Then in the concluding chapter of Srimad Bhagavad-Gita,
Vedavyasa again refers to Lord Krishna as Vasudeva in the Bhagavad-Gita,
chapter 18, verse 74:

ity aham vasudevasya parthasya ca mahatmanah
samvadam imam asrausam adbhutam roma-harsanam

Translation
Thus I have heard this wonderful divine, discourse between
two great souls Vasudeva and Arjuna and it is so electrifying that
my hair is standing on end.

Vasudeva is a primary expansion of the Supreme Lord Krishna
as revealed in the Vedic scriptures within the category designated as
catur-vyuha. The signifigance and paramount position of Vasudeva is
also thoroughly revealed in the illustrious Vedic scripture known as
the Srimad Bhagavatam, canto 1, chapter 2, verse 28 and 29,
as revealed below:

vasudeva para veda vasudeva para makhah
vasudeva para yoga vasudeva para kriyah
vasudeva param jnanam vasudeva param tapah
vasudeva paro dharmo vasudeva para gatih

Translation
The ultimate goal of all knowledge is the Supreme Lord Krishna
known as Vasudeva. The purpose of all austerities is to please
Vasudeva. The purpose of yoga is to realize Vasudeva.
All benefits which we receive are awarded by Vasudeva solely and
all penances are performed to learn Vasudeva onl　y.
Vasudeva is the Supreme knowledge.
Righteousness in its most complete form is rendering loving
devotional service unto Him.
Vasudeva is the supreme goal and ultimate reality of all existence.

So from this brief introduction,
the criteria of Vasudeva has been amply established .

http://www.bhagavad-gita.org/Articles/vasudeva.html