Question: "What is hyper-Calvinism and is it biblical?"
A simple definition is this: hyper-Calvinism is the belief that God saves the elect through His sovereign will with little or no use of the methods of bringing about salvation (such as evangelism, preaching, and prayer for the lost). To an unbiblical fault, the hyper-Calvinist over-emphasizes God's sovereignty and under-emphasizes man's responsibility in the work of salvation.
An obvious ramification of hyper-Calvinism is that it suppresses any desire to evangelize the lost. Most churches or denominations that hold to hyper-Calvinistic theology are marked by fatalism, coldness, and a lack of assurance of faith. There is little emphasis upon God's love for the lost and His own people but rather an unbiblical preoccupation with God's sovereignty, His election of the saved, and His wrath for the lost. The gospel of the hyper-Calvinist is a declaration of God's salvation of the elect and His damnation of the lost.
The Bible clearly teaches that God is sovereign over the entire universe (Daniel 4:34-35), including the salvation of men (Ephesians 1:3-12). But with God's sovereignty, the Bible also teaches that His motivation for saving the lost is love (Ephesians 1:4-5; John 3:16; 1 John 4:9-10) and that God's means of saving the lost is the proclamation of His Word (Romans 10:14-15). The Bible also declares that the Christian is to be passionate and determined in his/her sharing with unbelievers; as ambassadors for Christ, we are to "beg" people to be reconciled to God (2 Corinthians 5:20-21).
Hyper-Calvinism takes a biblical doctrine, Godís sovereignty, and pushes it to an unbiblical extreme. In doing so, the hyper-Calvinist downplays the love of God and the necessity of evangelism.