For mutual fund investing, KYC (know your customer) formalities have to be completed. You would need to submit PAN Card and Address proof (overseas address is acceptable). Once you are kyc compliant, you can invest into most of the mutual funds online or offline. Most of them have the facility to transact online.
You don't need dmat and trading account if you just want to be in mutual fund. It is required for trading in stocks, though you can also trade in mutual funds thru this if you want. IMO, it is cost effective to trade mutual fund thru trading acct as it will have low commissions cause some of the banks (like citibank, icici..) charge almost 2% as their own commission i.e. round trip of buy and sell cost 4% just as commission to middle bank.
happy investing.