All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, you should know that it is not permissible for a Muslim to open an account in banks which deal with Riba (interest and/or usury) or deal with such banks because this is considered as helping them in continuing their work which is based on Riba. Allaah Says (what means): {Help you one another in virtue, righteousness and piety but do not help one another in sin and transgression. And fear Allaah. Verily, Allaah is Severe in punishment.} [Quran 5:2].
Dealing with banks that deal with Riba without a necessity is helping in sin and transgression. Therefore, if all Muslims boycott these banks which deal with Riba and wage war against Allaah, then such banks will ultimately perish.
However, if a person is forced to open an account in a bank that deals with Riba, then it becomes permissible for him to do so; Allaah Says (what means): {While He has explained to you in detail what is forbidden to you, except under compulsion of necessity?}[Quran 6:119]. Nonetheless, he should deal with such a bank according to the necessity only. As regards a person who is not forced to open an account in such a bank, then it is not permissible at all for him to do so especially when Islamic banks are available.
As regards the "Takaful Family Scheme" that you inquired about in your question, it is a forbidden contract because it is based on risk and uncertainty. Indeed the bank may take from the account holder more than the amount that it will give him back at the time of his death or when he becomes handicapped. It might also be that the bank would take from him less than the amount they would give him, so this contract is based on risk and uncertainty, and so, it is forbidden.
To conclude, this scheme is one type of forbidden insurance, so it is forbidden for you to join it and it is not permissible for you to keep your account with it unless you are obliged to do so. You should always try to remember the saying of Allaah (which means): {And be afraid of the Day when you shall be brought back to Allaah. Then every person shall be paid what he earned, and they shall not be dealt with unjustly.}[Quran 2:281].
Allaah Knows best.