Creating a 10-Band Equalizer Using Web Audio API

By connecting every filter with the destination you are creating 5 paths (routes) so you will hear quintupling amplification of the source sound. It's not the correct way. You have to connect each filter in one line.

source.connect(filter1);
filter1.connect(filter2);
filter2.connect(filter3);
filter3.connect(filter4);
filter4.connect(filter5);
filter5.connect(context.destination);

As Matt D said, there should be no problem connecting the filters to the same destination.

I would however say that you probably want to use filters with type 5 (peaking), which lets all frequencies through and only amplifies/reduce at the frequency at which you've set the respective filter.frequency.value. That lets you connect the filters in series so you don't need 10 separate audio paths. You could also consider using a low-shelf filter as the first filter, and a hi-shelf filter as the tenth, which is rather common in equalizers. I can't remember if that's what winamp does, though.

Finally, if you go with the peaking filters in series, you don't need a separate gain node for each frequency, you just set the filter.gain.value for the specific filters.