There are two questions here:
- Is it possible that Voldemort's inability to get into Snape's mind is because the part of Voldemort's soul with a very powerful ability for legilimency was transferred to Harry?
The answer is no. Because, as Dumbledore tells Harry, splitting the soul leads to no decline in one's brain and magical powers:
"Never forget, though, that while his soul may be damaged beyond repair, his brain and his magical powers remain intact. It will take uncommon skill and power to kill a wizard like Voldemort even without his Horcruxes.”
- Half Blood Prince, Chapter 23, Horcruxes.
And Legilimency and Occlumency are magical powers where you channel your inherent magic and sometimes use the wand.
- Was Harry able to breach Snape's Occulumency in HBP because of the part of Voldemort's soul in him?
The answer is again no, as ash_k29 explains above. In the situation, Snape was teaching Harry Occlumency, trying to invade his mind. But then Harry unpredictably reacted with a Shield charm (Protego), which reflects (minor to moderate) spells, making Snape's Legilimens, which requires continued channeling of magic, rebound on himself. Of course, Snape would not have closed his mind as thoroughly as he does with Voldemort. How the intended victim is enabled the fruits of the rebounding Legilimens is not clear, but that something of the sort is to be expected is attested by the fact that Snape took special care to procure Dumbledore's Pensieve to remove some of his worst (and strategic, may be) memories.
While it is possible that Harry was endowed with some skill in Legilimency from the part of Voldemort' soul in him, like his ability for Parseltongue, there's nothing in canon to support this. And that Harry was never an accomplished Occlumens is attested by Dumbledore in the next book:
“Harry, you were never a good Occlumens —”
— Half Blood Prince, Chapter 25, The Seer Overheard.
As Pat Dobson points out above, Legilimency seems an acquired skill rather than an innate ability like the capacity to speak Parseltongue, and is thus less likely to be transfered in this way, from the manner in which the sinister process it is depicted in the books — Harry does not get any of Voldemort's skills in magic.