I recently created a version of Snake, the classic game popularized by Nokia, and I am planning on using it as an example of my game development skills. I just want to know if there are any copyright issues that could come up if I do this?
I am not a lawyer. Do not take this as legal advice.
There are three realms of intellectual property protection. Copyright, Trademark, and Patent.
You specifically asked for copyright in this question. Copyright protects the actual code, the actual art resources (including music), the actual wording in the instructions, and the actual runnable executable.
It is unlikely you are infringing upon any of these things.
A notable bit to be aware of when reimplementing games and copyright is that somehow The Tetris Company makes copyright infringement claims which noted infringement on the shapes of the blocks and the play field dimensions. I'm not a lawyer, and don't fully understand the grounds for that - just something to avoid and note that just because its not the things that are typically copyrightable that someone won't try to go after you. Another example of this can be found in the Triple Town / Yeti Town suit.
Lest this worry you that Nokia will come down and try to fuss about things, the game Snake is a rather old one. I played a game called Snake Byte in the days of the 6502 - a video of a remake of the game I played. That this game is so old and redone so many times, that you likely don't have much concern about reimplementing it again yourself - there's not one company that is holding on to the property and continue to gain from it.
Trademark is a minefield in IP. The use of the name 'snake' in reference to a game was trademarked - trademark search: name snake; owner nokia for computer games. The key thing there is the disclaimer: "NO CLAIM IS MADE TO THE EXCLUSIVE RIGHT TO USE "SNAKES" APART FROM THE MARK AS SHOWN" - you should be ok there too. Its also the only trademark that Nokia claims on Snakes. Note also that trademark is still active. Trademarking the word would likely have been problematic as lots of things use the word 'snake' as part of a game.
Patent is also a minefield. Typically, however you won't typically find the patent for a video game. On the other hand, there are numerous patents for board games. If you were trying to reimplenet a board game, say Khet You would run into patent 7,264,242 (which was upheld recently against a big name company). That said, trawling expired patents and deciphering the legalese back into rules can find some unpublished games that might be something to implement (5,145,182 was filed in 1990, expired in 2010, and had the patent fee lapse in 2000).
... as an example of my game development skills
In most creative fields, ignoring the legal rights and creating an unsellable version of a work strictly for portfolio development is perfectly fine. Especially if you're attempting to demonstrate your proficiency in some other area, such as with either graphic design or programming, rather than actual game design.
The story changes if you're attempting to demonstrate actual game design skills, which would require some actual creativity on top of "snake" that would keep the project form being mere plagiarism. And the story changes again if you want to actually distribute the game, such as in a smartphone app-store.
Placing your game development portfolio online is probably OK, especially if it's not a for-profit site and have sufficient contact information for someone who wants to present a IP claim so you can comply and simply take down the offending project.