If you wish to you can use your own HTML signup form. To do this navigate in the merchant panel to Configuration > Affiliate signup > HTML form (tab). There you can download a raw html code of the signup form.
The code gives you a basic idea of how the signup form should look like. You can customize the form the way you need, but you have to keep the given input field names.
Actually, the HTML form inherits  the field settings applied at Configuration > Affiliate Signup > Fields (tab).
You can find more info about html signup form here:

The signup form code does not include payout options, so if you want to manually add it there, you have to find out what IDs are assigned to each payout option and add the code manually.

How to do that?
First, you have to find out the Payout Option ID of the available payout options. In order to do that, assign the different payout options to certain affiliates (Affiliates > Affiliate manager > edit an affiliate > Payouts (tab) > set some payout option) and then export the list of affiliates at Tools > Import/Export > Export. The export file includes payout option IDs - that is what you need.

In the export file, find the affiliates you assigned a certain payout method to. The payout option ID can be found in the PAYOUTOPTIONID column.

Secondly you need the codes and names of the fields of the particular Payout Option.
In the merchant panel navigate to Configuration > Payout Options and click 'edit' (pencil icon) next to a payout option, e.g. "Bank/Wire transfer".
In the next screen you will see what fields the particular payout option includes. So there you find both the code and name of each field of the particular payout option.
You can find the same information also in the exported list of affiliates.

Now, in the HTML signup form you will have to create an input field with the value of particular payoutoptionid:
<input type="hidden" value="8444af30" name="payoutoptionid" />

Value 8444af30 was found in the  PAYOUTOPTIONID column in the exported list of affiliates.

Then you have to define input field(s) for each of the field of the particular payout option:
<input name="pp_email" type="text">

That's it.

Further customization possibilities:

1. Assign affiliate to campaign at signup
- activate plugin
Signup affiliate automatically to campaign
- add an input field with name a_cid to your signup form. This field must contain id of the campaign to which the new affiliate will be signed up.
If you wish to use a custom login form, check it out here: