You have two methods you need to disclose to your potential client: 1) PM, 2) Software, SDLC. These two methods are separate but need to work collaboratively.
I'll answer to #1 and leave #2 to the more technical folks on this exchange.
Your client is looking for how you scoped the project in terms of what is it you will deliver at the end of the project as well as interim deliveries during the project. This is best exhibited as a project scope statement, often found in a project charter but also within your Project Management Plan (PMP) as well as an initial Work Breakdown Structure (WBS) and possibly an initial WBS Dictionary. This is the what.
Your client will want to see how you are scheduling the work, the phases you are proposing, the amount of time you will need to make the deliveries. This is best displayed as a high level schedule with milestones, such as a Gantt chart, but your client may want to see your preliminary schedule showing your WBS loaded across time. This is the when.
Your client will want to see the cost of the project. This is best delivered as your cost proposal with price but your client may also want to see how much components of the project cost, which is exhibited as part of the Performance Measurement Baseline (PMB) and within your WBS Dictionary. This depends, however, on your type of contract with your client.
Finally, within your PMP, your client will be interested in your management approach to manage and control the project, covering such topics as risk, quality, communications to stakeholders, your procurement approach for materials you are going to need, etc.
This is a lot of information you likely don't have all sorted out yet since you do not even have the job. So scale this information accordingly, i.e., show what you have at a high level, more strategic kind of view. You will likely not have hard core plans and schedules to show at this stage so you will want to deliver this information more in a presentation format. The communication message is that you know how to organize, execute, manage and control work to get to the finish as efficiently as possible and to be able to communicate early and often when things go south.