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Foreigner living in the EU, best way to invest in US Stock Market?

like said above, interactive brokers should work for you. fees are cheap as well. when registering you have to ask permission for US stock exchanges. if you're looking to invest in S&P 500 you won't find SPX/SPY because of EU regulation, so look for other european ETFs that track the same index.
 
I would suggest to look into a Cyprus company setup as gains from trading titles such as stocks, bonds, ETF, Units is UCITs etc are exempt from taxation). As you are from Europe you can also combine this non-domiciled residency to achieve tax free dividends on personal income as well. For the trading platform Interactive Brokers or Saxo Bank are very good options to explore.