Should a Christian buy or invest in Bitcoins? The answer to this question is found in the Bible. Alternatively, one can do as Cornelius of old who devoted himself to God in prayer to the point where God revealed Himself to this man. The account of Cornelius is in the book of Acts chapter 10. According to verse 3, the angel of God was distinctly visible and audible to Cornelius and therefore couldn’t mistake the message for anything close to it.
The Bible is a guide for believers such that many refer to it as a manual for human beings. I believe that every little decision we take should be taken through the guidance of the Bible and that of the Holy Spirit who lives within us. It does not necessarily mean that you have to page the Bible each time you want to make the decision but it does mean you must enquire in your spirit if that which you want to do is in line with what the Bible says. It also calls for obedience to the Holy Spirit who lives within us.
Word of caution
God likes it for people to invest their money for a multiple return as detailed in the parable of talents (Matthew 25:14- 30). Jesus teaches that a person who hides his talent is not doing the right thing. Note that Jesus talks about investing with bankers to earn interest. However, the Bible condemns the love for money and calls for all believers to stay away from it (Hebrews 13:5). God has committed to never live you nor forsake you such that money should not drive you to believe in sustaining yourself through it.
These two verses seem to be having contrasting views yet these verses form part of your answer to the question. The Bible is rich with verses to guide you in taking such decisions. As long as you can quote scriptures that support what you have decided on and have peace within yourself about it. Last of all, I believe you are covered if the devil cannot accuse you before God (Revelation 12:10) for having his property in your possession.
You have to take a personal decision when you are convinced that the decision to participate in Bitcoins is approved of God.
After investing in Bitcoin
Christians should pay tax from all their proceeds. If you have earned some money in any way, including Bitcoins, you must declare your earnings with the responsible government entity and pay tax. If you are convinced that it is clean money that you earn from bitcoin, you must pay tithes.